Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94560 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7352 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1835 | C22 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ #1835 ◁ G | |
| LECS | G ▷ #1835 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 26 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)(3,13,15,44,16,45,48,64)(4,18,21,51)(6,12,27,42)(7,28,30,41)(8,31,32,46,34,59,38,14)(9,25,36,23)(11,26,40,20)(17,49,50,60,29,35,24,57)(19,43,47,62,53,33,58,61)(22,52,56,39)(37,55,63,54);(1,3)(2,8)(4,19)(5,24)(6,16)(7,29)(9,35)(10,38)(11,43)(12,34)(13,18)(14,23)(15,26)(17,20)(21,31)(22,53)(25,58)(27,50)(28,33)(30,48)(32,42)(36,60)(37,49)(39,44)(40,61)(41,62)(45,52)(46,54)(47,55)(51,64)(56,59)(57,63)Polycyclic presentation:
2^9|a^2=d,c^2=gi,d^2=f,e^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cg,d^b=degh,e^b=eh,f^b=fi,g^b=gi,d^c=dghi,e^d=ehi,g^d=gi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1756);