Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1490 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)(3,6)(4,7)(5,8)(11,12)(15,16)(19,23)(22,24)(25,26,29,31)(27,30,32,28);(1,3)(2,4)(5,7)(6,8)(9,10,14,13)(11,15,12,16)(25,26)(27,28)(29,31)(30,32);(1,3)(2,4)(5,7)(6,8)(9,11,14,12)(10,15,13,16)(17,18,21,20)(19,22,23,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,13,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,30)(26,27,31,32);(1,4,5,6)(2,7,8,3)(9,13,14,10)(11,16,12,15)(17,20,21,18)(19,24,23,22)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=i,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4359873);