Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31187 |
| 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 | |
1918 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)(5,7)(6,8)(9,10)(11,12,16,19,17,22,15,21)(13,18,23,34,26,31,25,33)(14,27,30,36,29,35,28,20)(24,38,39,41,40,42,37,32);(3,5,7,4)(6,9,10,8)(11,13)(12,18)(14,24)(15,25)(16,23)(17,26)(19,34)(20,32)(21,33)(22,31)(27,38)(28,37)(29,40)(30,39)(35,42)(36,41);(3,6,7,10)(4,8,5,9)(12,19)(13,23)(15,16)(18,31)(20,35)(21,22)(24,37)(25,26)(27,36)(28,30)(32,41)(39,40);(4,5)(8,9)(11,14,17,29)(12,20,22,36)(13,24,26,40)(15,30,16,28)(18,32,31,41)(19,35,21,27)(23,37,25,39)(33,38,34,42)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410863);