Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
| direct product | C4×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C42 | #121 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#389 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4);(5,6,10,14,11,15,9,25)(7,16,18,12,20,27,19,28)(8,21,23,29,24,35,22,13)(17,26,31,34,33,36,32,30);(5,7,10,18,11,20,9,19)(6,12,14,27,15,28,25,16)(8,17,23,31,24,33,22,32)(13,26,21,34,29,36,35,30);(5,8,9,22,11,24,10,23)(6,13,14,21,15,29,25,35)(7,17,19,32,20,33,18,31)(12,26,27,34,28,36,16,30)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261882);