Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#7338 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1316 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | #379 | C23 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1658 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 1658 conjugacy classes of subgroups — too many to draw. Only its 82 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);(3,4,6,10)(5,11,12,18,13,8,14,16)(7,9,15,17);(3,5)(4,8)(6,14)(7,13)(9,11)(10,16)(12,15)(17,18)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#149761, #72238404, #72238407, … |
| maximal normal subgroup of | C2×#149761, #72238404, #72238407, … |
| maximal quotient of | #72291972, #72340257 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56723);