Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#1554 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1100 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);(5,6,9,12,10,13,8,17)(7,14,15,19,16,11,18,20)(21,22)(23,29)(24,32)(25,27)(26,30)(28,34)(31,36)(33,35);(5,7,10,16)(6,11,13,14)(8,15,9,18)(12,19,17,20)(21,23,25,31)(22,26,27,35)(24,33,34,30)(28,36,32,29);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(22,27)(26,35)(28,32)(29,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89978);