Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52738 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5292 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);(4,9)(5,7)(12,15)(13,17)(19,20)(21,25)(22,24)(23,26)(27,28)(29,32)(30,31)(33,34);(3,4,8,9)(5,6,7,10)(12,15)(16,18)(20,22)(21,27)(26,30)(29,33);(3,5,8,7)(4,10,9,6)(11,12)(13,16)(14,15)(17,18)(19,21,24,27)(20,25,22,28)(23,29,31,33)(26,32,30,34);(3,6,8,10)(4,5,9,7)(11,13)(12,16)(14,17)(15,18)(19,22,24,20)(21,28,27,25)(23,30,31,26)(29,34,33,32);(3,7,8,5)(4,10,9,6)(19,23,24,31)(20,26,22,30)(21,29,27,33)(25,32,28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999793);