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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2474 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,14,16,15)(18,21)(19,23)(25,26)(27,30)(28,32)(29,31);(1,3,5,8)(2,4,6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,24)(20,21)(22,23)(25,27)(26,28)(29,32)(30,31);(2,6)(4,7)(10,13)(14,15)(17,18,20,21)(19,22,23,24)(25,27,29,32)(26,30,31,28);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,28)(26,27)(29,30)(31,32);(1,4,5,7)(2,3,6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=i,e^2=fh,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.
| maximal subgroup of | #72260758, #72260839, #72260841, … |
| maximal normal subgroup of | #72260758, #72260839, #72260841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 957063);