Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#8122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1460 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)(5,12,14,9)(6,16)(7,10)(8,11)(13,17,15,18)(19,20,23,27,24,28,22,26)(21,29,32,33,31,25,30,34);(3,5,7,15,8,14,6,13)(4,9,10,17,11,12,16,18)(19,21)(20,25,28,29)(22,30)(23,32)(24,31)(26,33,27,34);(3,6,8,7)(4,10,11,16)(5,13,14,15)(9,17,12,18)(19,22,24,23)(20,26,28,27)(21,30,31,32)(25,33,29,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88592);