Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1414 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,30,20,25)(21,32,31,26)(23,29,24,28);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,20)(18,25,27,30)(21,23,31,24)(26,29,32,28);(10,14)(11,16)(17,20,24,21,22,19,23,31)(18,26,29,30,27,32,28,25);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,24,19,22,31,23,20)(18,25,29,26,27,30,28,32);(2,5)(6,8)(10,14)(13,15)(18,27)(25,30)(26,32)(28,29)Polycyclic presentation:
2^9|a^2=ghi,b^2=i,c^2=h,d^2=fg,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7510317);