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×#31421 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1928 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)(5,15)(6,20)(7,23)(8,26)(9,14)(10,18)(11,22)(12,25)(13,28)(16,31)(17,33)(19,29)(21,34)(24,32)(27,30);(3,5,8,27,9,18,28,19)(4,10,13,30,14,15,26,29)(6,21,22,32,12,16,23,33)(7,24,25,34,11,17,20,31);(3,6,9,12)(4,11,14,7)(5,16)(8,23,28,22)(10,17)(13,25,26,20)(15,24)(18,21)(19,33)(27,32)(29,31)(30,34);(3,7,9,11)(4,12,14,6)(5,17,18,24)(8,20,28,25)(10,21,15,16)(13,22,26,23)(19,31,27,34)(29,32,30,33)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411870);