Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#11896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1066 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)(13,24,26,40,28,34,27,38)(14,29,31,20,33,39,32,19)(18,30,35,25,37,41,36,42);(3,5)(4,7)(6,10)(8,9)(11,13,17,28)(12,18,23,37)(14,30,33,41)(15,27,16,26)(19,38,20,40)(21,36,22,35)(24,29,34,39)(25,31,42,32);(5,10)(7,9)(11,14,15,31,17,33,16,32)(12,19,22,39,23,20,21,29)(13,25,26,41,28,42,27,30)(18,34,36,40,37,24,35,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389279);