Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573671 |
| 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 | C2×H(5,𝔽2) | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2878 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)(11,18,22,24)(12,16,13,23)(15,20,19,21);(1,2,6,4)(3,8,5,7)(9,11,14,22)(10,15,17,19)(12,21,13,20)(16,24,23,18);(1,3,6,5)(2,7,4,8)(9,10,14,17)(11,19,22,15)(12,16,13,23)(18,20,24,21);(1,4,6,2)(3,7,5,8)(9,10)(11,15)(12,23)(13,16)(14,17)(18,21)(19,22)(20,24);(1,5,6,3)(2,8,4,7)(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6532782);