Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#21733 |
| 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 | C24 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1834 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)(5,10)(6,8)(7,9)(11,12,17,22,18,23,16,33)(13,19,27,35,28,36,26,41)(14,29,15,21,31,38,30,20)(24,39,32,37,40,42,25,34);(3,5,7,6)(4,8,9,10)(11,13)(12,19)(14,24)(15,32)(16,26)(17,27)(18,28)(20,34)(21,37)(22,35)(23,36)(25,30)(29,39)(31,40)(33,41)(38,42);(11,14,16,30,18,31,17,15)(12,20,22,29,23,21,33,38)(13,24,26,25,28,40,27,32)(19,34,35,39,36,37,41,42);(3,6,7,5)(4,8,9,10)(11,15,17,31,18,30,16,14)(12,21,22,38,23,20,33,29)(13,25,27,24,28,32,26,40)(19,34,35,39,36,37,41,42)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281042);