Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,11)(12,16)(17,18,22,27,23,28,20,25)(19,29,30,26,31,32,21,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,15)(14,16)(17,19,23,31)(18,24,28,26)(20,30,22,21)(25,29,27,32);(2,6)(5,7)(9,11,13,10)(12,15,16,14)(17,20,23,22)(18,25,28,27)(19,30,31,21)(24,29,26,32);(3,8)(5,7)(10,11)(14,15)(17,21,22,19,23,30,20,31)(18,26,27,32,28,24,25,29);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=hi,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104681);