Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1532 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)(3,8)(4,7)(5,6)(9,10)(11,13)(12,16)(14,15)(18,21)(22,24)(25,26,28,30)(27,31,32,29);(2,5)(7,8)(10,15)(12,13)(19,23)(22,24)(25,26,28,30)(27,29,32,31);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,15)(14,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32);(1,4,6,3)(2,8,5,7)(9,12,14,13)(10,16,15,11)(17,19,20,23)(18,22,21,24)(25,27,28,32)(26,29,30,31);(1,5,6,2)(3,8,4,7)(9,13,14,12)(10,11,15,16)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926632);