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×#61655 |
| 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 | |
7568 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)(13,18)(14,25)(15,21)(16,29)(17,20)(19,26)(22,37)(23,34)(24,32)(27,36)(28,33)(30,35)(31,40)(38,42)(39,41);(3,5)(4,7)(6,10)(8,9)(11,13,17,23)(12,18,20,34)(14,22,27,38)(15,28,16,24)(19,31,35,41)(21,33,29,32)(25,37,36,42)(26,40,30,39);(11,14)(12,19)(13,22)(15,26)(16,30)(17,27)(18,31)(20,35)(21,36)(23,38)(24,39)(25,29)(28,40)(32,37)(33,42)(34,41);(5,10)(7,9)(12,20)(13,23)(19,35)(21,29)(22,38)(24,28)(25,36)(39,40);(4,8)(7,9)(11,15,17,16)(12,21,20,29)(13,24,23,28)(14,26,27,30)(18,32,34,33)(19,36,35,25)(22,39,38,40)(31,37,41,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484455);