Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11323 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(11,12,13,14)(16,19)(17,21)(20,22);(7,8)(9,10)(15,16,17,19,18,20,21,22);(3,4,5,6)(8,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#358973, C2×#359302, C2×#359042, … |
| maximal normal subgroup of | C2×#358973, C2×#359302, C2×#359042, … |
| maximal quotient of | C2×#121387, C2×#121341, C2×#121335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265625);