Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15258 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3037 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(10,13)(11,15)(17,18)(19,23)(20,21)(22,24)(25,26,28,31)(27,30,32,29);(2,4)(3,7)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,21)(22,24)(25,27)(26,29)(28,32)(30,31);(2,4)(6,8)(10,13)(11,15)(18,21)(22,23)(25,26,28,31)(27,29,32,30);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)Polycyclic presentation:
2^9|a^2=ghi,b^2=fghi,c^2=fgh,d^2=fi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238479, #72238480, #72238508, … |
| maximal normal subgroup of | #72238479, #72238480, #72238508, … |
| maximal quotient of | #72318356 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5026809);