Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C2×#62335 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6478 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)(5,9)(6,8)(7,10)(11,12,16,20)(13,17,22,31)(14,24,28,41)(15,29,26,19)(18,33,35,27)(21,36,25,40)(23,39,37,32)(30,42,34,38);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,25)(15,23)(16,22)(18,34)(19,32)(20,31)(21,28)(24,40)(26,37)(27,42)(29,39)(30,35)(33,38)(36,41);(11,14,26,33,16,28,15,27)(12,18,29,41,20,35,19,24)(13,21,37,42,22,25,23,38)(17,30,39,40,31,34,32,36);(5,10)(7,9)(11,13)(12,17)(14,21)(15,23)(16,22)(18,30)(19,32)(20,31)(24,36)(25,28)(26,37)(27,38)(29,39)(33,42)(34,35)(40,41);(4,8)(7,9)(13,22)(17,31)(21,25)(23,37)(30,34)(32,39)(36,40)(38,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10484823);