Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | C2×#13541 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #481 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
594 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);(3,4,7,9)(5,10,6,8)(13,20)(16,21)(18,26)(22,23);(3,5,7,6)(4,8,9,10)(11,12,14,24,15,17,25,19)(13,21,23,18,20,16,22,26);(3,6,7,5)(4,8,9,10)(12,16)(13,22)(14,25)(17,21)(18,24)(19,26)(20,23)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406759);