Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C22×#1287 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2);(3,4);(5,6,8,13,10,14,9,12)(7,15,16,11,18,19,17,20)(21,22)(23,29,31,35)(24,27)(25,28)(26,30,33,36)(32,34);(5,7,9,17,10,18,8,16)(6,11,13,19,14,20,12,15)(21,23)(22,26,28,33)(24,30)(25,31)(27,29,34,35)(32,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88383);