Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C22×C8 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | #8316 |
| prime radical | #3529 | C4 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 3 generators on 64 points:
(1,2,5,11,6,12,24,36,7,9,22,30,25,31,49,56)(3,13,16,38,17,39,45,61,14,37,41,54,42,60,62,64)(4,18,20,35,21,46,48,59,19,10,23,32,47,33,50,57)(8,26,28,51,29,52,55,15,27,43,34,44,53,63,58,40);(1,3)(2,8)(4,15)(5,16)(6,17)(7,14)(9,27)(10,13)(11,34)(12,29)(18,37)(19,40)(20,43)(21,44)(22,41)(23,26)(24,45)(25,42)(28,30)(31,53)(32,54)(33,39)(35,38)(36,58)(46,60)(47,51)(48,63)(49,62)(50,52)(55,56)(57,64)(59,61);(2,9)(3,14)(5,22)(12,31)(13,37)(15,40)(17,42)(20,23)(24,49)(28,34)(32,35)(39,60)(44,51)(48,50)(55,58)(57,59)Polycyclic presentation:
2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=ci,e^b=egi,f^b=fg,d^c=dg,f^c=fi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36788);