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 |
| prime radical | #62961 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,14,17)(11,18,21,24)(12,16,13,22)(15,23,19,20);(1,2,6,5)(3,8,4,7)(9,11,14,21)(10,15,17,19)(12,20,13,23)(16,24,22,18);(1,3,6,4)(2,7,5,8)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,20)(23,24);(1,4,6,3)(2,7,5,8)(9,10)(11,15)(12,22)(13,16)(14,17)(18,20)(19,21)(23,24);(1,5,6,2)(3,8,4,7)(9,12,14,13)(10,16,17,22)(11,20,21,23)(15,24,19,18)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479479);