Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,27)(20,31)(21,23)(22,29)(24,25)(26,28)(30,32);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,24)(26,29)(27,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,28,31,21,30,22,25)(18,24,29,32,23,20,26,27);(2,6)(5,7)(11,16)(13,15)(17,20)(18,25)(19,29)(21,24)(22,32)(23,31)(26,30)(27,28);(10,14)(13,15)(19,30)(20,24)(25,31)(27,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=gi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6816609);