Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32712 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#32795 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 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)(5,7)(6,8)(9,10)(11,12,16,19)(13,20,14,17)(15,18,23,26)(21,25,22,24)(27,28,31,35)(29,36,38,33)(30,39,32,34)(37,42,40,41);(3,5)(4,7)(6,9)(8,10)(11,13,16,14)(12,17,19,20)(15,22,23,21)(18,25,26,24)(29,37)(30,32)(33,41)(34,39)(36,42)(38,40);(4,8)(7,10)(11,14,16,13)(12,17,19,20)(15,21,23,22)(18,25,26,24)(27,29)(28,33)(30,37)(31,38)(32,40)(34,42)(35,36)(39,41);(5,9)(7,10)(11,15,16,23)(12,18,19,26)(13,21,14,22)(17,24,20,25)(27,30,31,32)(28,34,35,39)(29,37,38,40)(33,42,36,41)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6538413);