Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#24381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2847 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,6)(11,12,16,18,15,13,14,17);(4,6)(7,8)(9,10);(8,10)(11,13,15,12)(14,17,16,18);(4,6)116 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7517084, C2×#7514388, C2×#7516118, … |
| maximal normal subgroup of | C2×#7517084, C2×#7514388, C2×#7516118, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344924);