Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11915 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11915 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1533 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 888 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 18 points:
(1,2);(3,4,5,11,6,8,14,16,7,9,13,17,15,10,12,18);(4,8)(5,12)(6,15)(9,10)(11,17)(13,14);(4,9)(5,13)(8,10)(12,14)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#133259, C2×#483023, C2×#133275, … |
| maximal normal subgroup of | C2×#133259, C2×#483023, C2×#133275, … |
| maximal quotient of | #72219615 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390213);