Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#23720 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2180 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,15,21,17,20,16,22)(13,18,24,34,25,33,31,38)(14,26,28,36,30,37,29,19)(23,35,27,41,39,42,40,32);(4,7)(8,10)(11,13)(12,18)(14,27)(15,24)(16,31)(17,25)(19,35)(20,33)(21,34)(22,38)(23,29)(26,41)(28,39)(30,40)(32,37)(36,42);(11,14,17,30)(12,19,20,36)(13,23,25,39)(15,29,16,28)(18,32,33,41)(21,37,22,26)(24,40,31,27)(34,42,38,35);(3,5,6,9)(4,8,7,10)(12,20)(18,33)(19,36)(21,22)(26,37)(32,41)(34,38)(35,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339554);