Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#19051 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C2×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(4,7)(5,9)(8,10)(11,12,17,22)(13,23,26,35)(14,27,28,36)(15,21,16,20)(18,30,33,25)(19,31,34,29)(24,32,38,42)(37,39,41,40);(5,9)(7,10)(11,13)(12,18)(14,24)(15,30)(16,25)(17,26)(19,32)(20,35)(21,23)(22,33)(27,41)(28,38)(29,40)(31,39)(34,42)(36,37);(3,4)(5,10)(6,8)(7,9)(11,14)(12,19)(13,24)(15,29)(16,31)(17,28)(18,32)(20,27)(21,36)(22,34)(23,37)(25,39)(26,38)(30,40)(33,42)(35,41);(14,28)(19,34)(24,38)(27,36)(29,31)(32,42)(37,41)(39,40)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258650);