Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#25739 |
| 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 | |
5862 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)(7,8,11,13)(9,14,12,10);(3,4,6,5)(9,12)(10,14);(7,9,10,8,11,12,14,13);(3,5,6,4)86 conjugacy classes — view the full character table.
| maximal subgroup of | #72228199, C2×#6443328, C2×#502798, … |
| maximal normal subgroup of | #72228199, C2×#6443328, C2×#502798, … |
| maximal quotient of | #72228187 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350518);