Finite Group DatabasebetaFinite Group Databasebeta| order | 2048 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C210 · Schur multiplier (H2) = C244 |
| direct product | C24×2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C26 |
| commutator, Frattini, cosocle | C2 | C210 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
230005896 conjugacy classes of subgroups; only the 229755912 normal subgroups are shown.
This group has 230005896 conjugacy classes of subgroups — too many to draw. Only its 229755912 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 10 generators on 40 points:
(9,10,16,22)(11,23,27,17)(12,18,24,32)(13,19,31,35)(14,20,33,38)(15,21,34,28)(25,39,40,30)(26,29,37,36);(9,11,16,27)(10,17,22,23)(12,13,24,31)(14,25,33,40)(15,26,34,37)(18,35,32,19)(20,30,38,39)(21,36,28,29);(9,12)(10,18)(11,13)(14,28)(15,20)(16,24)(17,35)(19,23)(21,33)(22,32)(25,29)(26,30)(27,31)(34,38)(36,40)(37,39);(9,13)(10,19)(11,24)(12,27)(14,29)(15,30)(16,31)(17,18)(20,37)(21,25)(22,35)(23,32)(26,38)(28,40)(33,36)(34,39);(9,14)(10,20)(11,25)(12,28)(13,29)(15,18)(16,33)(17,30)(19,37)(21,24)(22,38)(23,39)(26,35)(27,40)(31,36)(32,34);(9,15)(10,21)(11,26)(12,20)(13,30)(14,32)(16,34)(17,36)(18,33)(19,25)(22,28)(23,29)(24,38)(27,37)(31,39)(35,40);(3,4);(5,6);(7,8);(1,2)(3,4)(5,6)(7,8)1040 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×2−1+8 |
| maximal normal subgroup of | C23×2−1+8 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((9,10,16,22)(11,23,27,17)(12,18,24,32)(13,19,31,35)(14,20,33,38)(15,21,34,28)(25,39,40,30)(26,29,37,36), (9,11,16,27)(10,17,22,23)(12,13,24,31)(14,25,33,40)(15,26,34,37)(18,35,32,19)(20,30,38,39)(21,36,28,29), (9,12)(10,18)(11,13)(14,28)(15,20)(16,24)(17,35)(19,23)(21,33)(22,32)(25,29)(26,30)(27,31)(34,38)(36,40)(37,39), (9,13)(10,19)(11,24)(12,27)(14,29)(15,30)(16,31)(17,18)(20,37)(21,25)(22,35)(23,32)(26,38)(28,40)(33,36)(34,39), (9,14)(10,20)(11,25)(12,28)(13,29)(15,18)(16,33)(17,30)(19,37)(21,24)(22,38)(23,39)(26,35)(27,40)(31,36)(32,34), (9,15)(10,21)(11,26)(12,20)(13,30)(14,32)(16,34)(17,36)(18,33)(19,25)(22,28)(23,29)(24,38)(27,37)(31,39)(35,40), (3,4), (5,6), (7,8), (1,2)(3,4)(5,6)(7,8));