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×#61649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7606 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)(11,12,17,22)(13,18,25,33)(14,26,28,36)(15,21,16,20)(19,30,34,27)(23,37,39,41)(24,35,29,32)(31,38,42,40);(3,5)(4,7)(6,10)(8,9)(11,13,17,25)(12,18,22,33)(14,23,28,39)(15,29,16,24)(19,31,34,42)(20,35,21,32)(26,37,36,41)(27,40,30,38);(11,14)(12,19)(13,23)(15,27)(16,30)(17,28)(18,31)(20,26)(21,36)(22,34)(24,38)(25,39)(29,40)(32,41)(33,42)(35,37);(5,10)(7,9)(11,15,17,16)(12,20,22,21)(13,24,25,29)(14,27,28,30)(18,32,33,35)(19,26,34,36)(23,38,39,40)(31,41,42,37);(4,8)(7,9)(13,25)(18,33)(23,39)(24,29)(31,42)(32,35)(37,41)(38,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484449);