Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25494 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1746 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);(3,4,7,10)(5,9,6,8)(11,12,17,31,15,20,16,21)(13,22,32,34,24,18,25,38)(14,26,30,19,28,37,29,36)(23,39,41,35,27,42,40,33);(3,5,7,6)(4,8,10,9)(11,13,17,32,15,24,16,25)(12,18,31,38,20,22,21,34)(14,27,30,40,28,23,29,41)(19,35,37,42,36,33,26,39);(11,14,15,28)(12,19,20,36)(13,23,24,27)(16,30,17,29)(18,33,22,35)(21,37,31,26)(25,41,32,40)(34,39,38,42);(3,6,7,5)(4,9,10,8)(12,20)(18,22)(19,36)(21,31)(26,37)(33,35)(34,38)(39,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6353325);