Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8440 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #379 |
| prime radical | C2×#3481 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,7,9,6,5,10,8)(12,15,16,17);(7,10)(8,9)(11,12)(13,17)(14,16)(15,18);(3,5)(4,6)(7,8)(9,10)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#186594, C2×#186596, C2×#186562, … |
| maximal normal subgroup of | C2×#186594, C2×#186596, C2×#186562, … |
| maximal quotient of | #72239478, C4×#8440 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93755);