Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C8×#102 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C24 | C4×D4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(7,8,9,11,10,12,13,14);(3,4,5,6)(15,16)(17,19)(18,21)(20,22);(16,19)(17,22)(20,21)224 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C4×C8×D8, C2×C8×#442 |
| maximal normal subgroup of | C2×C4×C8×D8, C2×C8×#442 |
| maximal quotient of | C2×C8×#357, C2×#128666, C2×C8×D4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89592);