Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C210 · Schur multiplier (H2) = C244 |
| direct product | C22×2−1+8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C28 |
| commutator, Frattini, cosocle | C2 | C210 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
229893860 conjugacy classes of subgroups; only the 229755616 normal subgroups are shown.
This group has 229893860 conjugacy classes of subgroups — too many to draw. Only its 229755616 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 10 generators on 68 points:
(5,6,13,20)(7,21,26,14)(8,15,27,42)(9,16,23,36)(10,17,35,40)(11,18,37,48)(12,19,38,32)(22,49,51,39)(24,50,55,34)(25,33,47,41)(28,43,52,62)(29,44,56,63)(30,45,57,65)(31,46,58,59)(53,67,68,61)(54,60,66,64);(5,7,13,26)(6,14,20,21)(8,22,27,51)(9,10,23,35)(11,24,37,55)(12,25,38,47)(15,39,42,49)(16,40,36,17)(18,34,48,50)(19,41,32,33)(28,29,52,56)(30,53,57,68)(31,54,58,66)(43,63,62,44)(45,61,65,67)(46,64,59,60);(8,27)(15,42)(22,51)(28,52)(29,56)(30,57)(31,58)(39,49)(43,62)(44,63)(45,65)(46,59)(53,68)(54,66)(60,64)(61,67);(5,8)(6,15)(7,22)(9,28)(10,29)(11,30)(12,31)(13,27)(14,39)(16,43)(17,44)(18,45)(19,46)(20,42)(21,49)(23,52)(24,53)(25,54)(26,51)(32,59)(33,60)(34,61)(35,56)(36,62)(37,57)(38,58)(40,63)(41,64)(47,66)(48,65)(50,67)(55,68);(5,9)(6,16)(7,10)(8,28)(11,32)(12,18)(13,23)(14,40)(15,43)(17,21)(19,37)(20,36)(22,29)(24,33)(25,34)(26,35)(27,52)(30,59)(31,45)(38,48)(39,63)(41,55)(42,62)(44,49)(46,57)(47,50)(51,56)(53,60)(54,61)(58,65)(64,68)(66,67);(5,10)(6,17)(7,23)(8,29)(9,26)(11,33)(12,34)(13,35)(14,16)(15,44)(18,47)(19,24)(20,40)(21,36)(22,52)(25,48)(27,56)(28,51)(30,60)(31,61)(32,55)(37,41)(38,50)(39,43)(42,63)(45,66)(46,53)(49,62)(54,65)(57,64)(58,67)(59,68);(5,11)(6,18)(7,24)(8,30)(9,32)(10,33)(12,16)(13,37)(14,34)(15,45)(17,47)(19,23)(20,48)(21,50)(22,53)(25,40)(26,55)(27,57)(28,59)(29,60)(31,43)(35,41)(36,38)(39,61)(42,65)(44,66)(46,52)(49,67)(51,68)(54,63)(56,64)(58,62);(5,12)(6,19)(7,25)(8,31)(9,18)(10,34)(11,36)(13,38)(14,41)(15,46)(16,37)(17,24)(20,32)(21,33)(22,54)(23,48)(26,47)(27,58)(28,45)(29,61)(30,62)(35,50)(39,64)(40,55)(42,59)(43,57)(44,53)(49,60)(51,66)(52,65)(56,67)(63,68);(3,4);(1,2)(3,4)1028 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×2−1+10, C22×#72258201, C23×2−1+8 |
| maximal normal subgroup of | C2×2−1+10, C22×#72258201, C23×2−1+8 |
| maximal quotient of | C23×2−1+8 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6,13,20)(7,21,26,14)(8,15,27,42)(9,16,23,36)(10,17,35,40)(11,18,37,48)(12,19,38,32)(22,49,51,39)(24,50,55,34)(25,33,47,41)(28,43,52,62)(29,44,56,63)(30,45,57,65)(31,46,58,59)(53,67,68,61)(54,60,66,64), (5,7,13,26)(6,14,20,21)(8,22,27,51)(9,10,23,35)(11,24,37,55)(12,25,38,47)(15,39,42,49)(16,40,36,17)(18,34,48,50)(19,41,32,33)(28,29,52,56)(30,53,57,68)(31,54,58,66)(43,63,62,44)(45,61,65,67)(46,64,59,60), (8,27)(15,42)(22,51)(28,52)(29,56)(30,57)(31,58)(39,49)(43,62)(44,63)(45,65)(46,59)(53,68)(54,66)(60,64)(61,67), (5,8)(6,15)(7,22)(9,28)(10,29)(11,30)(12,31)(13,27)(14,39)(16,43)(17,44)(18,45)(19,46)(20,42)(21,49)(23,52)(24,53)(25,54)(26,51)(32,59)(33,60)(34,61)(35,56)(36,62)(37,57)(38,58)(40,63)(41,64)(47,66)(48,65)(50,67)(55,68), (5,9)(6,16)(7,10)(8,28)(11,32)(12,18)(13,23)(14,40)(15,43)(17,21)(19,37)(20,36)(22,29)(24,33)(25,34)(26,35)(27,52)(30,59)(31,45)(38,48)(39,63)(41,55)(42,62)(44,49)(46,57)(47,50)(51,56)(53,60)(54,61)(58,65)(64,68)(66,67), (5,10)(6,17)(7,23)(8,29)(9,26)(11,33)(12,34)(13,35)(14,16)(15,44)(18,47)(19,24)(20,40)(21,36)(22,52)(25,48)(27,56)(28,51)(30,60)(31,61)(32,55)(37,41)(38,50)(39,43)(42,63)(45,66)(46,53)(49,62)(54,65)(57,64)(58,67)(59,68), (5,11)(6,18)(7,24)(8,30)(9,32)(10,33)(12,16)(13,37)(14,34)(15,45)(17,47)(19,23)(20,48)(21,50)(22,53)(25,40)(26,55)(27,57)(28,59)(29,60)(31,43)(35,41)(36,38)(39,61)(42,65)(44,66)(46,52)(49,67)(51,68)(54,63)(56,64)(58,62), (5,12)(6,19)(7,25)(8,31)(9,18)(10,34)(11,36)(13,38)(14,41)(15,46)(16,37)(17,24)(20,32)(21,33)(22,54)(23,48)(26,47)(27,58)(28,45)(29,61)(30,62)(35,50)(39,64)(40,55)(42,59)(43,57)(44,53)(49,60)(51,66)(52,65)(56,67)(63,68), (3,4), (1,2)(3,4));