Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | #34937 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1835 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,28,22,27,31,32)(19,24,29,25,21,26,20,30);(2,5)(3,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,31)(21,22)(23,29)(25,28)(26,27)(30,32);(10,14)(11,16)(17,20)(18,25)(19,23)(21,31)(22,29)(24,32)(26,28)(27,30);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,22,19)(18,26,27,24)(20,31,29,23)(25,28,30,32);(2,5)(6,8)(10,14)(13,15)(19,21)(20,29)(24,26)(25,30)Polycyclic presentation:
2^9|a^2=h,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dfi,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316431);