Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628842 |
| 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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,27,29,24)(20,26,22,25)(28,31,30,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,23)(24,27)(25,26)(31,32);(10,14)(13,16)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,31,26,32);(18,25)(20,22)(23,26)(24,31)(27,32)(28,30);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,26,23,25)(19,28,29,30)(24,32,27,31)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^d=ei,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537032);