Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33745 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1372 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)(11,12,15,20)(13,22,26,37)(14,19,17,29)(16,32,31,39)(18,33,23,27)(21,30,35,24)(25,36,28,42)(34,40,38,41);(4,7)(8,10)(11,12)(13,23)(14,29)(15,20)(16,21)(17,19)(18,26)(22,33)(24,32)(25,34)(27,37)(28,38)(30,39)(31,35)(36,41)(40,42);(11,13,24,40,17,28,31,33,15,26,30,41,14,25,16,27)(12,18,32,36,19,34,35,37,20,23,39,42,29,38,21,22);(3,5,6,9)(4,8,7,10)(11,14,15,17)(12,19,20,29)(13,25,26,28)(16,30,31,24)(18,34,23,38)(21,32,35,39)(22,36,37,42)(27,41,33,40)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530435);