Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26852 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,24)(14,30)(15,34)(16,22)(17,23)(18,27)(19,32)(20,25)(21,35)(26,38)(28,37)(29,39)(31,40)(33,41)(36,42);(3,5,7,6)(4,8,9,10)(11,13,16,27,17,29,35,28)(12,18,21,24,23,37,22,39)(14,31,32,42,33,15,26,25)(19,34,30,36,38,40,41,20);(3,6,7,5)(4,8,9,10)(11,14,16,32,17,33,35,26)(12,19,21,30,23,38,22,41)(13,25,27,31,29,42,28,15)(18,36,24,40,37,20,39,34);(4,9)(8,10)(11,15,16,25,17,31,35,42)(12,20,21,34,23,36,22,40)(13,26,27,14,29,32,28,33)(18,19,24,30,37,38,39,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351156);