Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15412 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2432 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)(4,5)(6,8)(10,11)(12,16)(18,20)(19,23)(26,30)(28,32);(2,4)(6,7)(10,11)(14,15)(19,23)(22,24)(26,30)(27,31);(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24)(25,26)(27,28)(29,30)(31,32);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,28)(29,31)(30,32);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,24,23,22)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|c^2=fi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72238475, #72238479, #72266266, … |
| maximal normal subgroup of | #72238475, #72238479, #72266266, … |
| maximal quotient of | #72319096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1979987);