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 | |
1514 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)(9,10,12,13)(11,15,16,14)(17,18,21,23)(19,26,20,24)(22,25,28,30)(27,31,29,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(18,23)(19,27)(20,29)(22,28)(24,31)(26,32);(9,11,12,16)(10,14,13,15)(17,19,22,27,21,20,28,29)(18,24,25,32,23,26,30,31);(17,20,22,29,21,19,28,27)(18,24,25,32,23,26,30,31);(19,20)(24,26)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=ghi,d^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373808);