Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #585 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
633 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 633 conjugacy classes of subgroups — too many to draw. Only its 145 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,31)(4,21,24,37)(5,11,26,36)(6,13,7,12)(9,19,32,18)(10,27,34,23)(16,41,44,29)(17,40,46,57)(22,50,53,49)(25,38,28,35)(30,48,55,45)(33,51,58,52)(39,47,56,43)(42,54,63,60)(59,61,64,62);(1,3,7,19,8,20,6,18)(2,9,13,31,14,32,12,15)(4,16,27,47,24,44,23,43)(5,17,28,48,26,46,25,45)(10,29,37,56,34,41,21,39)(11,30,38,57,36,55,35,40)(22,42,52,62,53,63,51,61)(33,54,50,64,58,60,49,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,27)(8,24)(9,29)(11,33)(12,21)(13,37)(14,34)(15,39)(17,42)(18,43)(19,47)(20,44)(25,51)(26,53)(28,52)(30,54)(31,56)(32,41)(35,49)(36,58)(38,50)(40,59)(45,61)(46,63)(48,62)(55,60)(57,64);(1,5)(2,11)(3,17)(4,22)(6,25)(7,28)(8,26)(9,30)(10,33)(12,35)(13,38)(14,36)(15,40)(16,42)(18,45)(19,48)(20,46)(21,49)(23,51)(24,53)(27,52)(29,54)(31,57)(32,55)(34,58)(37,50)(39,59)(41,60)(43,61)(44,63)(47,62)(56,64)Polycyclic presentation:
2^8|a^2=h,b^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,f^c=fh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144226, #144227, #144242, … |
| maximal normal subgroup of | #144226, #144227, #144242, … |
| maximal quotient of | #347884, #347885, #347886, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25940);