Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#60696 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C23×D4 |
| prime radical | C2×#3061 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,16,20)(13,17,22,30)(14,24,27,39)(15,28,25,19)(18,32,33,26)(21,34,36,42)(23,38,35,31)(29,40,41,37);(4,7)(8,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,29)(19,31)(20,30)(24,34)(25,35)(26,37)(27,36)(28,38)(32,40)(33,41)(39,42);(11,14,25,32,16,27,15,26)(12,18,28,39,20,33,19,24)(13,21,35,40,22,36,23,37)(17,29,38,42,30,41,31,34);(3,5,6,9)(4,8,7,10);(13,22)(17,30)(21,36)(23,35)(29,41)(31,38)(34,42)(37,40)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482510);