Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,31,36)(27,37,38,47)(28,33,43,49)(29,35,30,34)(32,40,45,41)(39,51,52,56)(42,50,44,48)(46,54,55,53);(1,2)(3,7)(4,8)(5,6)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(27,38)(32,45)(37,47)(39,52)(40,41)(46,55)(51,56)(53,54);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,27,30,41,31,38,29,40)(26,32,35,37,36,45,34,47)(28,39,42,53,43,52,44,54)(33,46,48,51,49,55,50,56);(1,4,5,3)(2,7,6,8)(11,18)(14,17)(19,24)(22,23)(25,28,31,43)(26,33,36,49)(27,39,38,52)(29,44,30,42)(32,46,45,55)(34,50,35,48)(37,51,47,56)(40,54,41,53);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,34,36,35)(27,40,38,41)(28,42,43,44)(32,47,45,37)(33,48,49,50)(39,53,52,54)(46,51,55,56)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=fg,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7094162);