Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#25091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2431 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4)(5,6)(7,8,10,9)(13,20)(17,25)(21,26)(22,24);(4,6)(7,9,10,8)(11,12)(13,21)(14,18)(15,19)(16,23)(17,22)(20,26)(24,25);(8,9)(11,13,15,20)(12,17,19,25)(14,22,16,24)(18,26,23,21);(4,6)(11,14,15,16)(12,18,19,23)(13,22,20,24)(17,26,25,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345941);