Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#39762 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,8,5)(6,9,7,10)(13,17)(15,18)(19,20)(21,26)(22,30)(23,28)(24,25)(27,32)(29,31)(33,34);(3,5,8,4)(6,9,7,10)(11,12,14,16)(13,15,17,18)(20,25)(23,26)(27,30)(31,33);(3,6,8,7)(4,9,5,10)(11,13,14,17)(12,15,16,18)(19,21,24,28)(20,26,25,23)(22,29,32,34)(27,33,30,31);(19,22,24,32)(20,27,25,30)(21,29,28,34)(23,31,26,33);(3,7,8,6)(4,10,5,9)(19,23,24,26)(20,28,25,21)(22,31,32,33)(27,34,30,29)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544632);