Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #34937 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1890 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)(9,10,13,15)(11,14,12,16)(17,18,22,20,24,26,23,21)(19,27,32,25,30,31,29,28);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,24,30)(18,25,26,28)(20,27,21,31)(22,29,23,32);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,26,24,21,23,18)(19,25,32,31,30,28,29,27);(1,4,5,3)(2,8,6,7)(17,21,23,26,24,20,22,18)(19,28,29,31,30,25,32,27);(17,22,24,23)(18,20,26,21)(19,29,30,32)(25,27,28,31)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365754);