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×#61204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7564 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,15,20)(13,19,23,17)(14,25,26,36)(16,29,28,21)(18,30,34,27)(22,37,38,42)(24,32,40,35)(31,39,33,41);(3,5)(4,7)(6,10)(8,9)(11,13,15,23)(12,17,20,19)(14,22,26,38)(16,24,28,40)(18,31,34,33)(21,32,29,35)(25,42,36,37)(27,39,30,41);(11,14,15,26)(12,18,20,34)(13,22,23,38)(16,30,28,27)(17,31,19,33)(21,36,29,25)(24,41,40,39)(32,37,35,42);(5,10)(7,9)(11,13)(12,19)(14,22)(15,23)(16,24)(17,20)(18,33)(21,35)(25,37)(26,38)(27,39)(28,40)(29,32)(30,41)(31,34)(36,42);(13,23)(17,19)(22,38)(24,40)(31,33)(32,35)(37,42)(39,41)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10483559);