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×#60720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ 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)(13,17,22,31)(14,24,27,41)(15,28,29,19)(18,33,35,26)(21,36,25,40)(23,39,38,32)(30,42,34,37);(4,7)(8,10)(11,13)(12,17)(14,25)(15,23)(16,22)(18,34)(19,32)(20,31)(21,27)(24,40)(26,42)(28,39)(29,38)(30,35)(33,37)(36,41);(11,14,15,26,16,27,29,33)(12,18,19,24,20,35,28,41)(13,21,23,37,22,25,38,42)(17,30,32,36,31,34,39,40);(3,5,6,9)(4,8,7,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,30)(19,32)(20,31)(24,36)(25,27)(26,37)(28,39)(29,38)(33,42)(34,35)(40,41);(13,22)(17,31)(21,25)(23,38)(30,34)(32,39)(36,40)(37,42)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482541);