Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9007 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #386 |
| prime radical | C2×#3150 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,8,10,7,5,6,9)(11,12)(13,22,25,33)(14,26)(15,20)(16,30)(17,21)(18,24,34,29)(19,27)(23,38,41,32)(28,35)(31,36)(37,39,42,40);(3,5,7,4)(6,9,8,10)(11,13,16,24,17,25,15,29)(12,18,30,22,21,34,20,33)(14,23,31,39,28,41,27,40)(19,32,26,37,36,38,35,42);(6,8)(9,10)(11,14)(12,19)(13,23)(15,27)(16,31)(17,28)(18,32)(20,35)(21,36)(22,37)(24,39)(25,41)(26,30)(29,40)(33,42)(34,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101803);