Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1861 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(18,24)(19,29)(20,21)(28,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,27)(20,26)(21,25)(22,24)(23,31)(28,30)(29,32);(9,12,13,11)(10,14,15,16)(17,19)(18,25)(20,23)(21,30)(22,29)(24,26)(27,31)(28,32);(2,6)(5,7)(17,20)(18,26)(19,23)(21,22)(24,25)(27,31)(28,32)(29,30);(3,8)(5,7)(17,21,22,20)(18,27,24,32)(19,30,29,23)(25,31,26,28)Polycyclic presentation:
2^9|b^2=fg,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597744);