Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2359 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,20,24,22)(25,26,29,30)(27,31,32,28);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,15,12)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(2,6)(3,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,24)(18,22,23,20)(25,28,29,31)(26,27,30,32);(1,4,5,7)(2,3,6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fhi,d^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4611811);