Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1864 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10,12,13)(11,14,15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,28)(29,31)(30,32);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,19,22,29)(18,23,26,32)(20,28,21,30)(24,31,25,27);(2,7)(6,8)(10,13)(14,16)(17,20,22,21)(18,24,26,25)(19,28,29,30)(23,31,32,27);(1,4,5,3)(2,8,7,6)(17,20,22,21)(18,25,26,24)(19,28,29,30)(23,27,32,31)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6561157);