Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#33791 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1646 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)(5,9)(6,8)(7,10)(12,18)(13,22)(14,17)(16,29)(19,33)(21,38)(23,35)(24,41)(25,40)(27,28)(30,31)(32,37)(34,42)(36,39);(3,5,6,10)(4,7,8,9)(11,12)(13,23)(14,26)(15,18)(16,21)(17,20)(19,25)(22,33)(24,34)(27,38)(28,32)(29,37)(30,42)(31,36)(35,40)(39,41);(5,10)(7,9)(11,13)(12,19)(14,24)(15,25)(16,30)(17,31)(18,23)(20,34)(21,39)(22,29)(26,36)(27,40)(28,41)(32,42)(33,38)(35,37);(4,8)(7,9)(11,14,15,17)(12,20,18,26)(13,24,25,31)(16,27,28,29)(19,34,23,36)(21,37,32,38)(22,30,40,41)(33,39,35,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530547);