Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422615 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15308 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,3,12,11)(4,15,18,13,19,26,17,8)(5,20,22,14,24,21,23,9)(16,25,30,28,31,27,29,32);(1,3)(2,7)(4,13,19,8)(5,21)(6,10)(9,23)(11,12)(14,22)(15,17,26,18)(16,28,31,32)(20,24)(25,29,27,30);(1,4,6,17,7,19,12,18)(2,8,10,26,3,13,11,15)(5,16,22,29,24,31,23,30)(9,25,20,32,14,27,21,28);(1,5,6,22,7,24,12,23)(2,9,10,20,3,14,11,21)(4,16,17,29,19,31,18,30)(8,25,26,32,13,27,15,28)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=e,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329812);