Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C16 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#13189 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C8 | C23 |
| commutator | C22 | C23×C16 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C24 | C22×C8 |
| prime radical | C23×D4 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,6)(8,10)(11,12,14,18,15,19,22,25,16,13,17,20,23,21,24,26);(3,4,5,6)(11,13,14,20,15,21,22,26,16,12,17,18,23,19,24,25);(7,8,9,10)224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402277);