Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8491 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8493 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(4,7,8,9)(11,12)(13,21,24,26)(14,17)(15,19)(16,20)(18,23,25,22);(3,4)(5,9)(6,8)(7,10)(12,17)(14,15)(18,21)(19,20)(22,23)(25,26);(5,10)(7,9)(11,13,16,24)(12,18,20,25)(14,23,15,22)(17,21,19,26)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94588);