Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
1746 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)(3,7)(4,6)(5,8)(10,13)(11,15)(18,22)(20,23)(25,26,31,30)(27,28,32,29);(1,2,4,6)(3,5,8,7)(9,10)(11,16)(12,13)(14,15)(17,18,21,22)(19,23,24,20)(25,27,31,32)(26,29,30,28);(3,8)(5,7)(10,13)(14,16)(18,22)(20,23)(25,28,31,29)(26,27,30,32);(1,2)(3,7)(4,6)(5,8)(9,11,12,15)(10,14,13,16)(17,19,21,24)(18,20,22,23)(25,29,31,28)(26,32,30,27);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,13)(14,15)(17,20)(18,19)(21,23)(22,24)(25,30,31,26)(27,29,32,28)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=g,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4035679);