Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C210 |
| direct product | C2×#10969 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
6994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(3,4)(5,6)(11,12,13,14)(15,16)(17,19)(18,21)(20,22);(7,8)(9,10)(16,19)(17,22)(20,21);(4,6)(8,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#393915, C2×#393909, C2×#357433, … |
| maximal normal subgroup of | C2×#393915, C2×#393909, C2×#357433, … |
| maximal quotient of | C2×#147621, C2×#147622, #72319916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264626);