Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 14 points:
(1,2);(4,5)(7,8)(9,12)(10,13)(11,14);(3,4,6,5)(7,8)(9,13)(10,12)(11,14);(8,12)(9,10)(13,14);(3,5,6,4)86 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6442607, #72286003, C2×#6442605, … |
| maximal normal subgroup of | C2×#6442607, #72286003, C2×#6442605, … |
| maximal quotient of | C2×#394048, C2×#261347, #72309172 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349794);