Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #45404 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1607 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,32,23,20,24,27)(19,28,21,26,30,25,29,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,25,20,28)(21,24,29,22)(26,32,31,27);(9,11,12,16)(10,13,14,15)(17,20,23,18)(19,25,30,28)(21,26,29,31)(22,32,24,27);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,29)(18,26,20,31)(19,22,30,24)(25,27,28,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,27,20,32)(19,29,30,21)(25,31,28,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6980179);