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) = C29 |
| direct product | C2×#8316 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C23 | C23×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #379 |
| prime radical | C22×H(5,𝔽2) | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3830 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)(5,9,10,7)(6,8)(11,12,13,15,14,16,17,18);(3,5)(4,7)(6,10)(8,9);(3,4)(5,7)(6,8)(9,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240223, #72240224, C2×#184911, … |
| maximal normal subgroup of | #72240223, #72240224, C2×#184911, … |
| maximal quotient of | C2×#129599, #72260930, C2×#129615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92099);