Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9758 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
| direct product | C2×#9758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #390 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1056 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);(4,7)(5,9)(8,10)(11,12)(13,17,22,32)(14,24)(15,28)(16,19)(18,26)(20,29)(21,35,25,39)(23,38,37,31)(27,40)(30,41,33,36)(34,42);(3,4,5,7,6,8,9,10)(11,13)(12,17,19,32)(14,25)(15,23)(16,22)(18,33,34,30)(20,31,28,38)(21,27)(24,39,40,35)(26,41)(29,37)(36,42);(4,8)(7,10)(11,14,15,26,16,27,29,42)(12,18,20,24,19,34,28,40)(13,21,23,36,22,25,37,41)(17,30,31,35,32,33,38,39)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112735);