Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| 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×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,18,13,16,12,17)(11,19,21,15,22,24,20,23);(1,2,6,5)(3,8,4,7)(9,11)(10,15)(12,21)(13,22)(14,20)(16,23)(17,24)(18,19);(1,3,6,4)(2,7,5,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(1,4,6,3)(2,7,5,8)(10,16)(15,23)(17,18)(19,24);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,f^b=fi,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6480596);