Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#21295 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2062 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,20,17,21,15,31)(13,18,25,34,23,33,24,40)(14,26,29,36,30,42,28,19)(22,37,39,35,27,41,38,32);(4,7)(8,10)(11,13,16,25,17,23,15,24)(12,18,20,34,21,33,31,40)(14,27,29,38,30,22,28,39)(19,35,26,41,36,32,42,37);(11,14,15,28,17,30,16,29)(12,19,20,26,21,36,31,42)(13,22,24,38,23,27,25,39)(18,32,34,37,33,35,40,41);(3,5,6,9)(4,8,7,10)(13,23)(18,33)(22,27)(24,25)(32,35)(34,40)(37,41)(38,39)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279344);