Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(18,22)(23,25)(24,29)(30,31);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,18,20,22)(19,24,26,29)(21,23,28,25)(27,30,32,31);(10,14)(13,15)(18,23)(21,28)(22,25)(24,30)(27,32)(29,31);(1,4,5,3)(2,6,7,8)(17,19,20,26)(18,24,22,29)(21,27,28,32)(23,30,25,31);(19,26)(24,29)(27,32)(30,31)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=fg,h^2=i,b^a=bf,c^a=cg,c^b=ch,h^b=hi,h^c=hi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339876);