Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,14,12)(11,13,16,15)(18,20)(19,23)(27,28)(30,31);(2,6)(4,7)(9,11,14,16)(10,15,12,13)(19,23)(22,24)(25,26,29,32)(27,30,28,31);(2,6)(4,7)(9,10,14,12)(11,13,16,15)(17,18,21,20)(19,22,23,24)(25,27,29,28)(26,30,32,31);(1,3,5,8)(2,4,6,7)(9,12,14,10)(11,15,16,13)(17,19,21,23)(18,22,20,24)(25,26)(27,30)(28,31)(29,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,15)(10,11,12,16)(17,20,21,18)(19,24,23,22)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=fg,d^2=hi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3875358);