Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #93313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1281 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | #1726 | C22 |
| socle | C2 | #7018 |
| prime radical | C23 | H(3,ℤ/4) |
| LCS | G ▷ C2×C8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ #1726 ▷ C22×C4 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,5,10)(3,14,16,42,18,44,51,58)(4,19,21,37)(6,11,27,40)(7,12,26,39)(8,28,29,48,31,53,59,64)(9,25,33,23)(13,41,38,20)(15,30,47,60,49,32,50,36)(17,43,24,55,46,45,56,61)(22,35,54,62)(34,57,63,52);(1,3)(2,8,11,30,12,31,13,32)(4,15)(5,24,26,56)(6,17)(7,18)(9,14,34,43,19,44,35,45)(10,36,38,59,39,60,40,29)(16,27,51,41)(20,46)(21,53,25,28)(22,48)(23,49)(33,61,62,58,37,55,63,42)(47,54,50,57)(52,64)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=h,e^2=f,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,d^b=dg,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 509);