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×#28937 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2238 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)(13,23)(14,27)(15,18)(16,21)(17,22)(19,24)(20,28)(25,39)(26,32)(29,36)(30,33)(31,34)(35,40)(37,38)(41,42);(3,5,7,6)(4,8,9,10)(12,18,22,21)(13,24,26,31)(19,34)(20,33,36,27)(23,32)(25,38,40,41)(35,39)(37,42);(3,6,7,5)(4,8,9,10)(11,13,15,24,17,26,16,31)(12,19,21,23,22,34,18,32)(14,25,28,38,30,40,29,41)(20,35,27,37,36,39,33,42);(4,9)(8,10)(11,14,15,28,17,30,16,29)(12,20,21,27,22,36,18,33)(13,25,24,38,26,40,31,41)(19,35,23,37,34,39,32,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6367237);