Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×#7182 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #102 |
| commutator | C8 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | D4⋊C8 |
| prime radical | C24 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4)(5,6)(7,8,10,16,12,17,20,14,13,18,21,15,11,22,9,19);(4,6)(8,14,17,15,18,19,22,16)(9,20)(10,21)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72210328, #72213035, C2×#143203, … |
| maximal normal subgroup of | #72210328, #72213035, C2×#143203, … |
| maximal quotient of | #72210317, #72210437, C2×#93386, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33988);