Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18,21,25)(19,26,27,32)(20,23,29,24)(22,30,31,28);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(19,27)(22,31)(26,32)(28,30);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,20,28,21,27,29,30)(18,22,24,26,25,31,23,32);(1,4,5,3)(2,6,8,7)(11,15)(13,16);(17,20,21,29)(18,23,25,24)(19,28,27,30)(22,32,31,26)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365064);