Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7372 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #353 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C2×#382 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 90 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);(3,4,5,7,6,8,9,10)(11,12,14,19,16,21,13,22)(15,18,23,17,24,20,25,26);(11,13,15,23,16,14,24,25)(12,17,18,19,21,26,20,22)56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56823);