Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7034 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #95 |
| commutator | C23 | C2×C4×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C25 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(3,4)(5,9,10,7)(6,8)(12,15)(13,17)(19,20,21,23,22,24,25,26);(3,5)(4,7)(6,10)(8,9)(11,12,13,16,14,15,17,18)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2342);