Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#7324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1414 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(4,7)(8,9)(11,12,14,15)(13,16)(17,18);(3,4)(5,9,10,7)(6,8)(11,13,15,18,14,17,12,16)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238070 |
| maximal normal subgroup of | #72238070 |
| maximal quotient of | C2×#93464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35351);