Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60557 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7586 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)(5,8)(6,7)(9,10)(11,12)(13,18)(14,26)(15,20)(16,29)(17,21)(19,27)(22,36)(23,32)(24,33)(25,39)(28,34)(30,35)(31,38)(37,42)(40,41);(4,7)(8,10)(11,13)(12,18)(14,22)(15,24)(16,25)(17,23)(19,31)(20,33)(21,32)(26,36)(27,38)(28,37)(29,39)(30,40)(34,42)(35,41);(11,14)(12,19)(13,22)(15,27)(16,30)(17,28)(18,31)(20,34)(21,35)(23,37)(24,38)(25,40)(26,29)(32,41)(33,42)(36,39);(3,5,6,9)(4,8,7,10)(13,23)(18,32)(22,37)(24,25)(31,41)(33,39)(36,42)(38,40);(11,15,17,16)(12,20,21,29)(13,24,23,25)(14,27,28,30)(18,33,32,39)(19,34,35,26)(22,38,37,40)(31,42,41,36)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482283);