Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,17)(7,12)(8,20)(10,19)(11,13)(14,30)(15,24)(16,26)(18,28)(21,25)(22,29)(23,31)(27,33)(32,34);(3,5,13,15,7,16,8,21)(4,9,20,25,12,26,11,24)(6,14,22,31,18,32,19,33)(10,27,28,34,29,23,17,30);(3,6,7,18)(4,10,12,29)(5,14)(8,22,13,19)(9,23)(11,28,20,17)(15,33)(16,32)(21,31)(24,30)(25,34)(26,27);(4,11)(8,13)(9,24)(10,28)(12,20)(15,21)(17,29)(19,22)(23,30)(25,26)(27,34)(31,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411867);