Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,15)(14,16)(17,18,22,21)(19,24,23,20)(25,26,29,31)(27,30,28,32);(1,3,5,4)(2,6,7,8)(10,11)(14,15)(17,19,22,23)(18,20,21,24)(25,26)(27,32)(28,30)(29,31);(9,10,13,11)(12,14,16,15)(17,20)(18,23)(19,21)(22,24)(25,27,29,28)(26,30,31,32);(1,2)(3,6)(4,8)(5,7)(9,11,13,10)(12,14,16,15)(17,20)(18,19)(21,23)(22,24)(25,26)(27,30)(28,32)(29,31);(1,4,5,3)(2,6,7,8)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,23,20)(25,28,29,27)(26,30,31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=h,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802710);