Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
| direct product | C2×#10732 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
5254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8,9,10)(11,12,15,18)(13,16,17,14);(11,13,16,12,15,17,14,18);(4,6)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#394070, C2×#108194, C2×#112557, … |
| maximal normal subgroup of | C2×#394070, C2×#108194, C2×#112557, … |
| maximal quotient of | C2×#106601, C2×#106575, #72260978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260592);