Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(9,10,13,16)(11,14,12,15)(17,18,22,25)(19,26,27,32)(20,24,21,30)(23,29,31,28);(11,12)(14,15)(17,19,21,28,22,27,20,29)(18,23,24,26,25,31,30,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15)(19,27)(23,31)(26,32)(28,29);(9,12,13,11)(10,15,16,14)(19,27)(23,31)(26,32)(28,29);(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,32,31,26)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=ghi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375292);