Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#61676 |
| 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 | |
7536 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)(5,9)(6,8)(7,10)(11,12)(13,20)(14,27)(15,23)(16,22)(17,21)(18,26)(19,29)(24,39)(25,34)(28,42)(30,38)(31,37)(32,36)(33,40)(35,41);(3,5)(4,7)(6,10)(8,9)(11,13,16,26)(12,18,22,20)(14,28,30,24)(15,31,17,25)(19,35,36,33)(21,37,23,34)(27,39,38,42)(29,40,32,41);(11,14,16,30)(12,19,22,36)(13,24,26,28)(15,29,17,32)(18,33,20,35)(21,27,23,38)(25,40,31,41)(34,39,37,42);(5,10)(7,9)(11,13)(12,20)(14,24)(15,31)(16,26)(17,25)(18,22)(19,35)(21,34)(23,37)(27,39)(28,30)(29,41)(32,40)(33,36)(38,42);(4,8)(7,9)(11,15,16,17)(12,21,22,23)(13,25,26,31)(14,29,30,32)(18,34,20,37)(19,27,36,38)(24,40,28,41)(33,39,35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484477);