Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
| direct product | C2×#20722 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
6754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17,18,15)(14,16);(4,7)(8,10);(11,13)(12,15)(14,18)(16,17);(3,5,6,9)(4,8,7,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72231639, #72231653, C2×#185475, … |
| maximal normal subgroup of | #72231639, #72231653, C2×#185475, … |
| maximal quotient of | #72236632, C2×#493962, C2×#493998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278131);