Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61197 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7618 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)(14,27)(15,20)(16,21)(17,22)(19,28)(23,37)(24,32)(25,36)(26,33)(29,34)(30,35)(31,38)(39,42)(40,41);(3,5)(4,7)(6,10)(8,9)(11,13,17,26)(12,18,22,33)(14,23,29,40)(15,24,16,25)(19,31,35,42)(20,32,21,36)(27,37,34,41)(28,38,30,39);(11,14)(12,19)(13,23)(15,28)(16,30)(17,29)(18,31)(20,34)(21,27)(22,35)(24,38)(25,39)(26,40)(32,41)(33,42)(36,37);(5,10)(7,9)(11,15,17,16)(12,20,22,21)(13,24,26,25)(14,28,29,30)(18,32,33,36)(19,34,35,27)(23,38,40,39)(31,41,42,37);(11,15,17,16)(12,21,22,20)(13,25,26,24)(14,28,29,30)(18,32,33,36)(19,27,35,34)(23,39,40,38)(31,41,42,37)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483553);