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×#33708 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1694 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)(5,8)(6,7)(9,10)(11,12)(13,26)(14,19)(15,21)(16,35)(17,37)(18,25)(20,31)(22,27)(23,36)(24,34)(28,42)(29,38)(30,39)(32,41)(33,40);(4,7)(8,10)(11,13,27,32,15,30,36,33,18,31,17,29,34,28,16,14)(12,19,35,42,24,38,37,20,25,40,23,39,21,41,22,26);(3,5,6,9)(4,8,7,10)(11,14,16,28,34,29,17,31,18,33,36,30,15,32,27,13)(12,20,22,38,21,42,23,19,25,26,37,41,24,39,35,40);(11,15,18,34)(12,21,25,24)(13,28,31,30)(14,29,33,32)(16,27,36,17)(19,38,40,41)(20,39,26,42)(22,23,37,35)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530368);