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×#25317 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1984 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)(5,9)(6,7)(8,10)(11,12,16,20,15,19,29,36)(13,21,24,38,23,17,31,33)(14,25,28,35,27,37,30,18)(22,39,40,32,26,42,41,34);(4,7)(8,9)(11,13,16,24,15,23,29,31)(12,17,20,33,19,21,36,38)(14,26,28,41,27,22,30,40)(18,34,25,39,35,32,37,42);(11,14,15,27)(12,18,19,35)(13,22,23,26)(16,30,29,28)(17,32,21,34)(20,37,36,25)(24,41,31,40)(33,39,38,42);(3,5,6,10)(4,8,7,9)(12,19)(17,21)(18,35)(20,36)(25,37)(32,34)(33,38)(39,42)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355267);