Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(10,12)(14,15)(17,18,21,24)(19,20,23,22);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,23)(21,24)(25,26,29,28)(27,31,32,30);(9,11,13,16)(10,14,12,15)(17,19,21,23)(18,22,24,20)(25,27,29,32)(26,30,28,31);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26,29,28)(27,30,32,31);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,22)(18,23,24,19)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712043);