Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61402 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,8,4,7)(9,10,13,18,15,19,14,24)(11,20,17,23,22,12,21,16);(1,3,6,4)(2,7,5,8)(9,11,14,21,15,22,13,17)(10,16,18,20,19,23,24,12);(1,2,6,5)(3,8,4,7)(9,12,13,23,15,20,14,16)(10,17,24,22,19,21,18,11);(1,4,6,3)(2,7,5,8)(9,13,15,14)(10,18,19,24)(11,21,22,17)(12,23,20,16);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,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, 6461865);