Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1434 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)(9,10,13,12)(11,14,16,15)(17,18,23,27)(19,28,29,32)(20,30,31,24)(21,26,22,25);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,15)(13,16)(17,19,21,30,23,29,22,24)(18,20,25,32,27,31,26,28);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(17,20,23,31)(18,24,27,30)(19,26,29,25)(21,32,22,28);(2,6)(3,8)(11,16)(14,15)(17,21,23,22)(18,25,27,26)(19,24,29,30)(20,28,31,32);(2,6)(5,7)(17,19,21,30,23,29,22,24)(18,20,25,32,27,31,26,28)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=ghi,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7514631);