Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61181 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3489 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,18)(14,25)(15,21)(16,29)(17,20)(19,26)(22,37)(23,33)(24,32)(27,36)(28,34)(30,35)(31,39)(38,41)(40,42);(3,5)(4,7)(6,10)(8,9)(11,13,17,24)(12,18,20,32)(14,22,27,40)(15,28,16,23)(19,31,35,41)(21,34,29,33)(25,37,36,42)(26,39,30,38);(11,14)(12,19)(13,22)(15,26)(16,30)(17,27)(18,31)(20,35)(21,36)(23,38)(24,40)(25,29)(28,39)(32,41)(33,37)(34,42);(5,10)(7,9)(12,20)(18,32)(19,35)(21,29)(25,36)(31,41)(33,34)(37,42);(11,15,17,16)(12,21,20,29)(13,23,24,28)(14,26,27,30)(18,33,32,34)(19,36,35,25)(22,38,40,39)(31,37,41,42)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483505);