Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33610 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1986 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8)(9,18)(10,15)(11,26)(12,24)(13,17)(14,21)(16,19)(20,31)(22,37)(23,33)(25,30)(27,38)(28,35)(29,36)(32,34);(7,9,19,34,12,22,27,36,13,23,25,35,10,20,11,21)(8,14,26,31,15,28,30,33,17,29,38,37,24,32,16,18);(4,6)(7,8)(9,14)(10,24)(11,16)(12,15)(13,17)(18,21)(19,26)(20,32)(22,28)(23,29)(25,38)(27,30)(31,34)(33,36)(35,37);(7,10,13,12)(8,15,17,24)(9,20,23,22)(11,25,27,19)(14,28,29,32)(16,26,30,38)(18,31,33,37)(21,35,36,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530166);