Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #772778 |
| 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 | |
2055 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)(3,6)(4,7)(5,8)(10,14)(11,16)(17,18,21,24)(19,20,23,22)(25,26)(27,30)(28,32)(29,31);(1,3,4,8)(2,5,7,6)(9,10,13,14)(11,15,16,12)(25,26,29,31)(27,30,32,28);(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,22,24,20)(25,27,29,32)(26,30,31,28);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,15)(10,16,14,11)(17,20,21,22)(18,23,24,19)(25,26)(27,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 712052);