Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60711 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6738 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)(11,12,16,20)(13,21,22,34)(14,18,26,32)(15,28,23,19)(17,29,30,25)(24,35,36,42)(27,39,37,33)(31,40,41,38);(4,7)(8,10)(13,22)(17,30)(21,34)(24,36)(25,29)(31,41)(35,42)(38,40);(11,13,23,29,16,22,15,25)(12,17,28,34,20,30,19,21)(14,24,37,40,26,36,27,38)(18,31,39,42,32,41,33,35);(3,5,6,9)(4,8,7,10)(11,14)(12,18)(13,24)(15,27)(16,26)(17,31)(19,33)(20,32)(21,35)(22,36)(23,37)(25,38)(28,39)(29,40)(30,41)(34,42);(14,26)(18,32)(24,36)(27,37)(31,41)(33,39)(35,42)(38,40)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482530);