Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
| direct product | C2×#7132 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#95 | C23 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C22×#95 ◁ G | |
| LECS | G ▷ C2×#95 ▷ C24 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7,8,9)(11,12,13,15,14,16,17,18)(19,20)(21,24,26,23)(22,25);(3,4,6,8)(5,9,10,7)(20,23)(21,26)(24,25)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31316);