Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C4×D8 |
| prime radical | C24 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
614 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);(3,4)(5,8)(6,7)(9,10)(11,12,17,22,15,20,31,23)(13,21,26,35,25,18,16,32)(14,27,29,39,28,42,30,34)(19,36,38,41,37,33,40,24);(4,7)(8,10)(11,13,17,26,15,25,31,16)(12,18,22,32,20,21,23,35)(14,24,29,36,28,41,30,33)(19,34,38,27,37,39,40,42);(3,5,6,9)(4,8,7,10)(11,14,15,28)(12,19,20,37)(13,24,25,41)(16,33,26,36)(17,29,31,30)(18,34,21,39)(22,38,23,40)(27,35,42,32)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401990);