Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438967 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 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)(9,10)(11,16)(12,14)(13,15)(17,18,21,23)(19,26,28,32)(20,25,22,30)(24,29,31,27);(1,3,4,8)(2,5,6,7)(11,12)(14,16)(18,23)(24,31)(25,30)(26,32);(9,11,13,12)(10,14,15,16)(17,19,22,27,21,28,20,29)(18,24,25,26,23,31,30,32);(9,12,13,11)(10,14,15,16);(17,20,21,22)(18,25,23,30)(19,27,28,29)(24,32,31,26)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370465);