Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667460 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 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)(9,10,13,15)(11,14,12,16);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,18)(19,24)(20,29)(21,23)(22,27)(25,30)(26,31)(28,32);(9,12,13,11)(10,14,15,16)(18,23)(21,31)(24,32)(25,29)(26,27)(28,30);(2,6)(5,7)(17,19,22,20)(18,24,27,29)(21,28,31,30)(23,32,26,25);(3,8)(5,7)(17,20,22,19)(18,25,27,32)(21,28,31,30)(23,29,26,24)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6599952);