Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23912 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#24080 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3493 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3250 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)(13,17)(14,19)(15,18)(16,20)(21,24)(22,26)(23,25);(3,5,7,6)(4,8,9,10)(11,13)(12,17)(14,21)(15,23)(16,22)(18,25)(19,24)(20,26);(3,6,7,5)(4,8,9,10)(12,18)(14,15)(17,24)(19,20)(21,23)(25,26);(4,9)(8,10)(11,14,16,15)(12,19,20,18)(13,21,22,23)(17,24,26,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360131);