Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#10902 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4,6,5)(7,8,11,17,13,18,12,16)(9,19,22,30,23,29,21,14)(10,24,25,32,27,31,26,15)(20,34,36,38,37,28,35,33);(7,9,12,21,13,23,11,22)(8,14,16,29,18,30,17,19)(10,20,26,35,27,37,25,36)(15,28,31,38,32,34,24,33);(3,5,6,4)(7,10,11,25,13,27,12,26)(8,15,17,24,18,32,16,31)(9,20,22,36,23,37,21,35)(14,28,19,33,30,34,29,38)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261129);