Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#29510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(19,20,24,27)(21,25,29,32)(22,31,26,23)(28,34,33,30);(3,5,6,10)(4,7,8,9)(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,25,27,32)(22,28,26,33)(23,30,31,34);(12,15)(16,18)(19,22,23,20,24,26,31,27)(21,28,30,25,29,33,34,32);(4,8)(7,9)(11,13,14,17)(12,16,15,18)(21,29)(25,32)(28,33)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6368737);