Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,6,7)(5,8,9,10)(19,20,23,24)(21,27,29,25)(22,30,31,26)(28,32,34,33);(5,9)(8,10)(11,12,15,16)(13,14,17,18)(20,24)(22,31)(25,27)(28,34);(4,7)(8,10)(12,16)(13,17)(19,21,23,29)(20,25,24,27)(22,28,31,34)(26,32,30,33);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,17)(16,18)(19,20)(21,27)(22,26)(23,24)(25,29)(28,33)(30,31)(32,34);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,18)(16,17)(19,22,23,31)(20,26,24,30)(21,28,29,34)(25,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8009657);