Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
| direct product | C2×#7048 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | #95 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
686 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 686 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 512 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,15,17,16,13,14,18)(20,24,25,26);(4,7,9,8)(11,13,16,12)(14,17,15,18)(19,20,22,24,23,25,21,26)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2542);