Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#26853 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,10)(6,8)(7,9)(11,12)(13,15)(14,17)(16,21)(18,22)(19,23)(20,25)(24,26);(3,5,7,6)(4,8,9,10)(11,13,16,23,18,24,17,20)(12,19,14,15,22,25,21,26);(3,6,7,5)(4,8,9,10)(11,14,16,22,18,21,17,12)(13,19,23,15,24,25,20,26);(4,9)(8,10)(11,15,16,25,18,26,17,19)(12,20,14,13,22,23,21,24)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351154);