Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1258513 |
| 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 | #14762 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,12,15)(11,14,16,13)(17,18,21,24)(19,22,23,20)(25,26,29,31)(27,32,30,28);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,23)(18,22,24,20)(25,26)(27,32)(28,30)(29,31);(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,27,29,30)(26,28,31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,28)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,15)(14,16)(17,20,21,22)(18,23,24,19)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gh,c^2=ghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1194374);