Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24023 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3486 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4162 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);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,24)(15,27)(16,20)(18,26)(19,30)(21,38)(22,36)(23,33)(25,37)(28,32)(29,35)(31,40)(34,41)(39,42);(4,7)(8,10)(11,13,16,23)(12,17,20,33)(14,21,25,39)(15,28,30,22)(18,34,35,31)(19,36,27,32)(24,38,37,42)(26,41,29,40);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,21)(15,29)(16,25)(17,31)(19,37)(20,35)(22,41)(23,39)(24,27)(26,30)(28,40)(32,42)(33,34)(36,38);(4,7)(8,10)(14,25)(18,35)(21,39)(24,37)(26,29)(31,34)(38,42)(40,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347548);