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 |
| prime radical | #35078 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,30,21,25,19,23)(20,27,26,24,29,31,28,32);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,22)(18,23,25,30)(20,28,29,26)(24,27,32,31);(11,15)(13,16)(17,20,21,29)(18,24,25,32)(19,26,22,28)(23,31,30,27);(1,3,4,8)(2,6,5,7)(11,15)(13,16);(2,5)(6,7)(10,14)(13,16)(18,25)(23,30)(24,32)(27,31)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfhi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6542167);