Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52299 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1568 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)(11,13)(12,14)(15,16)(17,18,22,20,21,23,31,29)(19,25,28,27,26,24,32,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(18,23)(19,26)(20,29)(28,32);(3,8)(5,7)(10,14)(11,15)(17,19)(18,24)(20,30)(21,26)(22,28)(23,25)(27,29)(31,32);(2,6)(3,8)(10,14)(13,16)(19,26)(24,25)(27,30)(28,32);(17,20,22,23,21,29,31,18)(19,27,28,24,26,30,32,25)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=cfgi,d^b=dfi,e^b=ei,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315037);