Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(17,18,24,28)(19,22,27,23)(20,30,32,29)(21,26,25,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,28,24,27,23,18)(20,29,31,21,32,30,26,25);(10,13)(14,16)(17,20,22,31,24,32,23,26)(18,25,19,29,28,21,27,30);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,21)(18,26)(19,20)(22,30)(23,29)(24,25)(27,32)(28,31);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,22,24,23)(18,27,28,19)(20,31,32,26)(21,29,25,30)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466168);