Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,27,29,23)(20,25,22,26)(28,32,30,31);(1,3,5,8)(2,6,7,4)(9,11,12,16)(10,13,14,15)(19,28)(20,22)(23,31)(25,26)(27,32)(29,30);(10,14)(13,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,31)(26,32);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(18,24)(23,27)(25,26)(31,32);(17,20,21,22)(18,25,24,26)(19,28,29,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417634);