Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#30700 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1961 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);(3,4,6,8)(5,7,9,10)(11,12)(13,18)(14,23)(15,25)(16,21)(17,22)(19,24)(20,26);(3,4)(5,7)(6,8)(9,10)(11,13,17,15)(12,18,22,25)(14,20,19,21)(16,23,26,24);(3,5,6,9)(4,7,8,10)(11,14,17,19)(12,13,22,15)(16,25,26,18)(20,24,21,23);(4,8)(7,10)(11,15,17,13)(12,19,22,14)(16,23,26,24)(18,21,25,20)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408763);