Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#21382 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#3444 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(12,15,16,17);(11,12,13,17,14,16,18,15);(4,7)(8,10);(3,5,6,9)(4,8,7,10)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239190, #72239236, #72239292, … |
| maximal normal subgroup of | #72239190, #72239236, #72239292, … |
| maximal quotient of | #72239192, #72239495, C2×#207854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279565);