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 | #52139 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1569 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,23,21,22,25,32,31)(19,26,28,29,20,24,30,27);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,22)(21,27)(23,28)(25,26)(29,31)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(11,16)(17,20,22,19)(18,24,25,26)(21,27,31,29)(23,30,32,28);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,21,23,25,22,31,32,18)(19,27,28,26,20,29,30,24);(10,14)(13,15)(19,20)(24,26)(27,29)(28,30)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=efi,c^b=cgi,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, 6332836);