Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8863 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | #385 |
| prime radical | C25 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
970 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);(4,8)(5,6)(9,10)(11,12,15,19,16,20,26,32)(13,17,23,29,22,28,36,40)(14,18,25,31,24,30,37,41)(21,33,35,42,34,27,38,39);(3,4)(5,8)(6,10)(7,9)(11,13,15,23,16,22,26,36)(12,17,19,29,20,28,32,40)(14,21,25,35,24,34,37,38)(18,27,31,39,30,33,41,42);(3,5,7,6)(4,8,9,10)(11,14)(12,18)(13,21)(15,25)(16,24)(17,27)(19,31)(20,30)(22,34)(23,35)(26,37)(28,33)(29,39)(32,41)(36,38)(40,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99416);