Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1359 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)(9,10,12,14)(11,15,16,13)(17,18)(19,29)(20,27)(21,32)(22,28)(23,26)(24,31)(25,30);(1,3)(2,5)(4,8)(6,7)(17,19,22,31)(18,24,28,29)(20,21,30,23)(25,32,27,26);(9,11,12,16)(10,13,14,15)(17,20,23,19,22,30,21,31)(18,25,26,24,28,27,32,29);(2,6)(5,7)(17,21,22,23)(18,26,28,32)(19,30,31,20)(24,25,29,27);(3,8)(5,7)(17,20,21,31,22,30,23,19)(18,27,32,24,28,25,26,29)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=ghi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725028);