Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2523 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,22)(19,25)(24,29)(26,31);(3,8)(5,7)(10,14)(11,15)(17,19)(18,23)(20,25)(21,26)(22,28)(24,30)(27,31)(29,32);(2,6)(3,8)(10,14)(13,16)(18,22)(23,28)(24,29)(30,32);(19,25)(23,28)(26,31)(30,32)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=dfi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314658);