Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1936 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)(11,12,15,20,17,22,16,21)(13,18,24,33,25,34,31,38)(14,26,29,37,30,36,28,19)(23,35,27,41,40,42,39,32);(3,5,7,6)(4,8,9,10)(11,13)(12,18)(14,27)(15,24)(16,31)(17,25)(19,35)(20,33)(21,38)(22,34)(23,28)(26,41)(29,40)(30,39)(32,36)(37,42);(3,6,7,5)(4,8,9,10)(11,14)(12,19)(13,23)(15,28)(16,29)(17,30)(18,32)(20,36)(21,26)(22,37)(24,39)(25,40)(27,31)(33,42)(34,41)(35,38);(4,9)(8,10)(11,15,17,16)(12,20,22,21)(13,24,25,31)(14,28,30,29)(18,33,34,38)(19,36,37,26)(23,39,40,27)(32,42,41,35)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360374);