Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#3050 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1092 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)(5,9,10,7)(6,8)(11,12,15,21,17,23,16,22)(13,24,27,40,29,41,28,38)(14,30,32,20,34,39,33,19)(18,31,35,26,37,42,36,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,31)(15,28)(16,27)(17,29)(19,38)(20,40)(21,36)(22,35)(23,37)(24,39)(25,32)(26,33)(30,41)(34,42);(5,10)(7,9)(11,14,15,32,17,34,16,33)(12,19,22,39,23,20,21,30)(13,25,27,31,29,26,28,42)(18,24,36,38,37,41,35,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389284);