Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
| direct product | C2×#28060 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12,14,16)(13,15,17,18)(19,20,26,31)(21,28,32,34)(22,27,23,33)(24,30,25,29);(3,5,7,6)(4,8,9,10)(11,13)(12,15)(14,17)(16,18)(19,21,25,23,26,32,24,22)(20,27,30,34,31,33,29,28);(3,6,7,5)(4,8,9,10)(13,17)(15,18)(19,22,24,32,26,23,25,21)(20,28,29,33,31,34,30,27);(4,9)(8,10)(19,23,25,32,26,22,24,21)(20,28,30,27,31,34,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364700);