Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#7721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1190 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)(5,9,10,7)(6,8)(11,12)(13,20,22,25)(14,23)(15,18)(16,19)(17,21,26,24)(27,28,30,33,31,34,37,40)(29,32,36,39,35,38,41,42);(3,5)(4,7)(6,10)(8,9)(11,13,16,22)(12,17,19,26)(14,21,15,24)(18,25,23,20)(27,29)(28,32)(30,36)(31,35)(33,39)(34,38)(37,41)(40,42);(4,8)(7,9)(11,14,16,15)(12,18,19,23)(13,21,22,24)(17,25,26,20)(29,35)(32,38)(36,41)(39,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 68881);