Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #327 |
| commutator | C2×C8 | C2×C16 |
| Frattini, cosocle | C2×#346 | C22 |
| socle | C2 | #7102 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×#346 ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 64 points:
(1,2,5,11,7,14,30,46,6,13,29,45,26,10,28,40,8,12,31,44,32,41,58,63,24,39,56,59,4,22,25,43)(3,15,17,48,19,37,52,42,20,49,53,64,23,35,55,61,21,47,54,60,16,36,50,62,18,38,27,57,51,9,33,34);(1,3)(2,9)(4,23)(5,27)(6,20)(7,16)(8,21)(10,38)(11,42)(12,35)(13,37)(14,47)(15,41)(17,29)(18,24)(19,32)(22,49)(25,52)(26,51)(28,50)(30,55)(31,53)(33,58)(34,59)(36,39)(40,48)(43,60)(44,62)(45,61)(46,57)(54,56)(63,64)Polycyclic presentation:
2^9|a^2=d,c^2=fi,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cf,d^b=def,f^b=fi,g^b=gi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1087);