Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,24,21,25,20,29)(19,26,27,31,28,23,30,32);(1,2)(3,4)(5,7,8,6)(10,13)(14,16)(17,19)(18,23)(20,30)(21,28)(22,27)(24,32)(25,26)(29,31);(2,4)(6,7)(11,15)(14,16);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=dei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287341);