Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431723 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3487 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,17)(13,22)(14,18)(15,19)(20,24)(21,23);(2,5)(6,8)(12,13)(15,23)(17,22)(19,21);(9,11)(10,15)(12,19)(13,21)(14,20)(16,22)(17,24)(18,23);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,19)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,24,23,16)Polycyclic presentation:
2^9|d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,e^b=ei,g^b=gi,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339291);