Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533075 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2797 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(17,18,24,27,23,19,22,26)(20,25,29,31,21,28,30,32);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,23,18)(20,25,21,28)(22,27,24,26)(29,31,30,32);(2,6)(7,13)(8,14)(15,16)(17,20,23,21)(18,25,19,28)(22,29,24,30)(26,31,27,32);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21,23,20)(18,25,19,28)(22,29,24,30)(26,32,27,31);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,22,23,24)(18,26,19,27)(20,29,21,30)(25,31,28,32)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,f^b=fi,g^b=gi,f^c=fi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451398);