Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573715 |
| 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 | C24 | C25 |
| prime radical | #32926 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3165 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,7,4,8)(12,20)(14,17)(15,23)(18,21);(1,3,6,4)(2,7,5,8)(9,10,13,16)(11,17,19,14)(12,15,20,23)(18,24,21,22);(1,2,6,5)(3,8,4,7)(9,11,13,19)(10,14,16,17)(12,21,20,18)(15,24,23,22);(1,4,6,3)(2,8,5,7)(9,12,13,20)(10,15,16,23)(11,18,19,21)(14,22,17,24);(1,5,6,2)(3,7,4,8)(10,16)(12,20)(14,17)(18,21)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,d^2=h,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534452);