Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#27293 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2598 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)(5,8)(6,7)(9,10)(11,12)(13,24)(14,30)(15,34)(16,22)(17,23)(18,27)(19,26)(20,35)(21,31)(25,41)(28,38)(29,40)(32,37)(33,42)(36,39);(4,7)(8,10)(11,13,16,27,17,29,31,28)(12,18,21,24,23,38,22,40)(14,15,26,35,33,36,32,25)(19,39,30,20,37,34,42,41);(3,5,6,9)(4,8,7,10)(11,14,31,32,17,33,16,26)(12,19,22,42,23,37,21,30)(13,25,28,36,29,35,27,15)(18,20,40,39,38,41,24,34);(4,7)(8,10)(11,15,31,25,17,36,16,35)(12,20,22,39,23,41,21,34)(13,26,28,14,29,32,27,33)(18,37,40,30,38,19,24,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348034);