Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33697 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| socle | C23 | C22×D8 |
| prime radical | C2×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1714 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)(13,20)(14,19)(15,21)(16,23)(17,22)(18,25)(24,32)(26,38)(27,34)(28,39)(29,35)(30,37)(31,40)(33,41)(36,42);(4,7)(8,10)(11,13,17,27,15,26,31,30,18,28,33,42,32,29,16,14)(12,19,23,35,24,36,41,39,25,37,40,38,21,34,22,20);(3,5,6,9)(4,8,7,10)(11,14,16,29,32,42,33,28,18,30,31,26,15,27,17,13)(12,20,22,34,21,38,40,37,25,39,41,36,24,35,23,19);(11,15,18,32)(12,21,25,24)(13,26,28,29)(14,27,30,42)(16,17,31,33)(19,34,37,36)(20,38,39,35)(22,40,41,23)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530356);