Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33894 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,16,18)(13,21,22,39)(14,25,29,40)(15,30,27,20)(17,32,33,23)(19,36,35,28)(24,41,31,37)(26,38,42,34);(3,5)(4,7)(6,10)(8,9)(13,22)(17,33)(21,39)(23,32)(24,31)(26,42)(34,38)(37,41);(11,13,16,22)(12,17,18,33)(14,26,29,42)(15,31,27,24)(19,37,35,41)(20,38,30,34)(21,40,39,25)(23,36,32,28);(5,10)(7,9)(12,18)(17,33)(19,35)(20,30)(21,39)(25,40)(34,38)(37,41)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530902);