Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,34)(27,35,36,47)(28,32,42,50)(29,43,44,33)(31,38,45,39)(37,51,52,56)(40,49,41,48)(46,54,55,53);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,36)(31,45)(35,47)(37,52)(38,39)(46,55)(51,56)(53,54);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,39,30,36,44,38)(26,31,33,35,34,45,43,47)(28,37,41,53,42,52,40,54)(32,46,49,51,50,55,48,56);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,30,42)(26,32,34,50)(27,37,36,52)(29,41,44,40)(31,46,45,55)(33,49,43,48)(35,51,47,56)(38,54,39,53);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,44)(26,33,34,43)(27,38,36,39)(28,40,42,41)(31,47,45,35)(32,48,50,49)(37,53,52,54)(46,51,55,56)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fg,d^2=fi,e^2=fi,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961050);