Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45408 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1482 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)(3,6)(4,7)(5,8)(10,13)(11,15)(17,18,24,27,23,28,22,21)(19,29,31,25,20,26,32,30);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,16,15,14)(17,19,23,20)(18,25,28,30)(21,26,27,29)(22,31,24,32);(1,3,5,4)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,26,28,29)(21,25,27,30)(22,31,24,32);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,21,24,18,23,27,22,28)(19,30,31,29,20,25,32,26);(1,4,5,3)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,27,28,21)(19,31,20,32)(25,29,30,26)Polycyclic presentation:
2^9|a^2=fhi,b^2=gi,c^2=ghi,d^2=fgi,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7030782);