Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#32920 |
| 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 | |
5442 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);(4,6)(8,11)(9,13)(12,14);(3,4,5,6);(7,8)(9,14)(10,12)(11,13);(9,13)(11,14)98 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6628968, C2×#500507, C2×D4×#453, … |
| maximal normal subgroup of | C2×#6628968, C2×#500507, C2×D4×#453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536157);