Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#13106 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C8×D4 |
| prime radical | C24 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
638 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)(11,12,17,22,16,23,33,40,15,20,32,24,31,21,38,39)(13,14,27,29,26,30,37,42,25,18,34,35,41,28,19,36);(4,7)(8,10)(11,13,17,27,16,26,33,37,15,25,32,34,31,41,38,19)(12,18,22,35,23,28,40,36,20,14,24,29,21,30,39,42);(3,5,6,9)(4,8,7,10)(11,14,15,18)(12,19,20,37)(13,24,25,22)(16,30,31,28)(17,29,32,35)(21,34,23,27)(26,39,41,40)(33,42,38,36)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401968);