Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27808 |
| 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 | |
1962 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)(11,12,16,21,17,22,15,18)(13,23,24,33,26,35,31,19)(14,27,29,36,30,32,28,20)(25,34,38,37,39,42,41,40);(3,5,6,9)(4,7,8,10)(12,18,22,21)(13,24,26,31)(19,33)(20,32,36,27)(23,35)(25,38,39,41)(34,42)(37,40);(4,8)(7,10)(11,13)(12,19)(14,25)(15,24)(16,31)(17,26)(18,23)(20,34)(21,35)(22,33)(27,40)(28,38)(29,41)(30,39)(32,37)(36,42);(11,14,16,29,17,30,15,28)(12,20,18,32,22,36,21,27)(13,25,31,41,26,39,24,38)(19,34,23,37,33,42,35,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364897);