Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C2×#7962 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,8,9)(5,7,10,6)(11,12)(13,21,24,25)(14,18)(15,19)(16,20)(17,23,26,22);(3,5,7,9,8,10,6,4)(11,13)(12,17,20,26)(14,22)(15,23)(16,24)(18,25,19,21);(3,6,8,7)(4,5,9,10)(11,14,16,15)(12,18,20,19)(13,22,24,23)(17,25,26,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88368);