Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C27 |
| direct product | C2×#8851 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #385 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1506 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);(7,10)(11,12,15,13,14,16,18,17);(3,4)(5,7)(6,8)(9,10)(11,13,15,16,14,17,18,12);(3,5)(4,7)(6,9)(8,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72313358, #72313370, #72313374, … |
| maximal normal subgroup of | #72313358, #72313370, #72313374, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99222);