Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
| direct product | C2×#60319 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C2×C4 | C26 |
| socle | C23 | C24×C4 |
| prime radical | C2×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
6716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12,16,15,17,20,21,25)(13,18,14,19,22,24,23,26);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,16)(15,19)(17,22)(20,24)(21,23)(25,26);(5,10)(7,9)(11,14)(12,19)(13,21)(15,24)(16,22)(17,23)(18,25)(20,26);(4,8)(7,9)(13,22)(14,23)(18,24)(19,26);(11,15,16,20,17,25,21,12)(13,19,14,24,22,26,23,18)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10481675);