Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×#121 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(12,15)(17,22)(19,20)(21,23);(2,5)(6,8);(9,10,15,22,13,17,12,14)(11,16,18,23,20,24,19,21);(9,11)(10,16)(12,19)(13,20)(14,21)(15,18)(17,24)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,h^a=hi,h^d=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537550);