Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1634 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,15)(13,16)(17,18,24,27,23,19,22,28)(20,29,31,30,32,25,21,26);(2,6)(7,8)(11,12)(14,15)(17,19,23,18)(20,25,32,29)(21,26,31,30)(22,28,24,27);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,29)(21,22)(23,32)(24,31)(26,27)(28,30);(9,12,13,11)(10,15,16,14)(17,21,23,31)(18,26,19,30)(20,22,32,24)(25,27,29,28);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,22,23,24)(18,27,19,28)(20,31,32,21)(25,30,29,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=gh,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396606);