Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#23492 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C2×#120 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(4,5)(7,8)(9,13)(10,15)(11,19)(12,16)(14,17)(18,22)(20,21);(3,4)(5,6)(7,9)(8,13)(10,17)(11,20)(12,18)(14,15)(16,22)(19,21);(3,5,6,4)(7,9)(8,14)(10,17)(11,20)(12,18)(13,19)(15,22)(16,21);(7,10,12,11)(8,15,16,19)(9,17,18,20)(13,14,22,21)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339073);