Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#13057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×#454 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2118 conjugacy classes of subgroups; only the 258 normal subgroups are shown.
This group has 2118 conjugacy classes of subgroups — too many to draw. Only its 258 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 22 points:
(1,2);(4,6)(7,8,9,13,12,17,10,15)(11,20,19,22,21,16,18,14);(3,4,5,6)(8,13)(9,10)(14,16)(15,17)(18,19)(20,22);(7,9)(8,14)(10,12)(11,19)(13,20)(15,16)(17,22)(18,21)80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#494622, D3×#13057, #72261939, … |
| maximal normal subgroup of | C2×#494622, #72261939, #72261982, … |
| maximal quotient of | #72240575, #72240578, C2×#139785, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401810);