Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8697 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
984 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);(3,4,7,10,8,6,5,9)(11,12,16,19,17,14,15,25)(13,20,23,21,24,26,22,18);(3,5,8,7)(4,9,6,10)(11,13,16,23,17,24,15,22)(12,18,25,26,14,21,19,20);(3,6)(4,8)(5,10)(7,9)(11,14)(12,17)(13,21)(15,25)(16,19)(18,24)(20,23)(22,26)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96617);