Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7158 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | #340 |
| prime radical | C24 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12,15,19)(13,14,21,23)(16,18,24,26)(17,25,22,20);(11,13,16,20,17,18,23,19,15,21,24,25,22,26,14,12)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#129120, C2×#128692, C2×C8×#360, … |
| maximal normal subgroup of | C2×#129120, C2×#128692, C2×C8×#360, … |
| maximal quotient of | C2×#93270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33859);