Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8510 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
| direct product | C2×#8511 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2328 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)(12,17)(13,20)(14,15)(19,22)(21,23)(24,25);(3,4)(5,9)(6,8)(7,10)(11,12)(13,18)(14,22)(15,17)(16,19)(20,25)(21,26)(23,24);(5,10)(7,9)(11,13,16,21)(12,18,19,26)(14,20,15,23)(17,24,22,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94855);