Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23391 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23802 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4082 conjugacy classes of subgroups; only the 722 normal subgroups are shown.
This group has 4082 conjugacy classes of subgroups — too many to draw. Only its 722 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) = 256 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(7,8,10,11)(9,12,13,14)(15,16,17,21,19,22,18,20);(3,4)(5,6)(8,11)(12,14)(16,20)(17,18)(21,22);(4,6)(16,21)(17,18)(20,22);(7,9,10,13)(8,12,11,14)(16,22)(20,21)104 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#6432606, #72217716, C2×#6467359, … |
| maximal normal subgroup of | C2×#6432606, #72217716, C2×#6467359, … |
| maximal quotient of | C2×#359042, C2×#244406, C2×#244402, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339787);