Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2112 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)(3,8)(4,6)(5,7)(9,10,13,16)(11,12,15,14)(17,18,21,20)(19,22,24,23)(25,26)(27,28)(29,32)(30,31);(1,3)(2,4)(5,6)(7,8)(9,10,13,16)(11,12,15,14)(17,19,21,24)(18,22,20,23)(25,27)(26,30)(28,32)(29,31);(1,4,5,8)(2,6,7,3)(9,11,13,15)(10,14,16,12)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16);(1,4)(2,3)(5,8)(6,7)(9,12,13,14)(10,15,16,11)(17,20,21,18)(19,23,24,22)(25,28)(26,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 950901);