Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| 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 | |
1609 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)(17,18,23,26,22,19,21,25)(20,27,28,32,30,24,29,31);(2,6)(7,8)(11,12)(14,15)(17,19,22,18)(20,24,30,27)(21,25,23,26)(28,32,29,31);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,28)(25,32)(26,31);(9,12,13,11)(10,15,16,14)(17,21,22,23)(18,25,19,26)(20,28,30,29)(24,31,27,32);(1,4,5,3)(2,7,6,8)(10,16)(14,15)(17,21,22,23)(18,26,19,25)(20,28,30,29)(24,32,27,31)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,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6396580);