Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#11391 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | #379 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(11,12,15,20,16,19,28,33)(13,21,24,34,23,17,27,30)(14,18,26,31,25,32,38,42)(22,29,37,40,36,35,39,41);(3,4,6,8)(5,9,10,7)(11,13)(12,17)(14,22)(15,27)(16,23)(18,29)(19,21)(20,34)(24,28)(25,36)(26,39)(30,33)(31,41)(32,35)(37,38)(40,42);(3,5,6,10)(4,7,8,9)(11,14)(12,18)(13,22)(15,26)(16,25)(17,29)(19,32)(20,31)(21,35)(23,36)(24,37)(27,39)(28,38)(30,40)(33,42)(34,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265982);