Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33706 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1726 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)(13,25)(14,19)(15,33)(16,36)(17,37)(18,24)(20,27)(21,26)(22,35)(23,34)(28,40)(29,39)(30,38)(31,42)(32,41);(4,7)(8,10)(11,13,26,32,17,29,35,30,18,27,16,28,34,31,15,14)(12,19,33,42,23,40,36,20,24,38,22,39,37,41,21,25);(3,5,6,9)(4,8,7,10)(11,14,15,31,34,28,16,27,18,30,35,29,17,32,26,13)(12,20,21,40,37,42,22,19,24,25,36,41,23,39,33,38);(13,27)(14,30)(19,38)(20,25)(28,32)(29,31)(39,42)(40,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530366);