Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | |
1742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,14,12)(11,16,13,15)(18,21)(19,23)(25,26,28,29)(27,30,32,31);(1,2)(3,7)(4,8)(5,6)(9,11,14,13)(10,15,12,16)(18,21)(22,24)(26,29)(30,31);(2,6)(4,7)(9,10,14,12)(11,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,27,28,32)(26,30,29,31);(1,3,5,8)(2,4,6,7)(9,12,14,10)(11,15,13,16)(17,19,20,23)(18,22,21,24)(25,26)(27,30)(28,29)(31,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,16,12,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=hi,d^2=h,e^2=gh,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, 4038414);