Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | #16258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2251 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)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,19)(20,24)(25,26)(27,28)(29,31)(30,32);(2,4)(6,8)(9,10,12,14)(11,13,16,15)(18,19)(22,23)(25,27,29,32)(26,30,31,28);(3,7)(6,8)(11,16)(13,15)(17,18,21,19)(20,22,24,23)(25,28,29,30)(26,27,31,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,19,21,18)(20,23,24,22)(25,28)(26,27)(29,30)(31,32);(1,4,5,2)(3,8,7,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=h,d^2=i,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1923974);