Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9345 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#9354 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,8,9,7,10,6,5)(12,16)(13,19)(14,22)(18,24)(20,21)(23,25);(3,5,8,10,7,9,6,4)(11,12,14,16,15,18,22,24)(13,17,20,23,21,26,19,25);(3,6,7,8)(4,9,10,5)(11,13)(12,17)(14,20)(15,21)(16,23)(18,26)(19,22)(24,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107122);