Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#23363 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,13,15,14,16,17,18)(19,20)(21,28)(22,26)(23,32)(24,27)(25,31)(29,34)(30,33);(5,10)(7,9)(19,21,24,31)(20,25,27,28)(22,30,23,29)(26,34,32,33);(3,5,6,10)(4,7,8,9)(19,22,24,23)(20,26,27,32)(21,29,31,30)(25,33,28,34);(4,8)(7,9)(19,23,24,22)(20,26,27,32)(21,30,31,29)(25,33,28,34)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6314836);