Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1354 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)(4,8)(5,6)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(9,10,15,11)(12,13,14,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(2,6)(7,8)(9,11,14,16,15,10,12,13);(1,3,5,4)(2,7,6,8)(9,12,15,14)(10,13,11,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(1,4,5,3)(2,8,6,7)(9,13,14,11,15,16,12,10)(19,27)(22,25)(26,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=h,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,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, 6748827);