Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1414 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
| direct product | C2×#7500 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | #532 | C23 |
| socle | C22 | #1393 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
728 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4,5,10)(6,9,12,17)(7,11,13,18)(8,14,16,15);(4,8,11,9)(5,12,13,14)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72340530, #72340531, C2×#152748 |
| maximal normal subgroup of | #72340530, #72340531, C2×#152748 |
| maximal quotient of | C2×#94176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59037);