Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9427 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,10,8,12,6,11)(5,13,15,17,16,18,14,9)(19,20,25,28,24,29,23,27)(21,30,32,26,33,34,31,22);(3,5)(4,9,12,17)(6,14)(7,15)(8,16)(10,13,11,18)(19,21)(20,26)(22,29)(23,32)(24,33)(25,31)(27,34)(28,30);(4,10)(6,7)(9,13)(11,12)(14,15)(17,18)(19,22,24,26)(20,21,29,33)(23,30,25,34)(27,32,28,31)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108038);