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 | #34911 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1704 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,31)(20,24,21,25)(27,30,29,32);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(19,27)(20,21)(23,30)(24,25)(28,29)(31,32);(10,14)(13,15)(17,19,22,28)(18,23,26,31)(20,27,21,29)(24,30,25,32);(2,3)(6,7)(11,16)(13,15);(1,4,5,8)(2,6,3,7)(17,20,22,21)(18,24,26,25)(19,27,28,29)(23,30,31,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=fgi,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, 7419687);