Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#7867 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,13,16,14,15,17,18);(4,7,9,8);(12,15)(16,18)176 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#110273, #72209983, #72209984, … |
| maximal normal subgroup of | C2×#110273, #72209983, #72209984, … |
| maximal quotient of | C22×#7867, #72237307, C2×#96443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88093);