Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | |
1184 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)(27,35,36,52)(28,32,42,51)(29,43,44,33)(31,45,46,38)(37,53,39,56)(40,50,41,49)(47,54,48,55);(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)(31,46)(35,52)(37,39)(38,45)(47,48)(53,56)(54,55);(1,2,5,4)(3,6,8,7)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,38,30,36,44,45)(26,31,33,35,34,46,43,52)(28,39,41,54,42,37,40,55)(32,48,50,56,51,47,49,53);(1,2,5,4)(3,7,8,6)(12,20)(15,24)(17,19)(21,23)(25,28,30,42)(26,32,34,51)(27,37,36,39)(29,41,44,40)(31,47,46,48)(33,50,43,49)(35,53,52,56)(38,55,45,54);(1,4,5,2)(3,7,8,6)(10,16)(11,18)(15,24)(17,19)(25,29,30,44)(26,33,34,43)(27,38,36,45)(28,40,42,41)(31,35,46,52)(32,49,51,50)(37,54,39,55)(47,56,48,53)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=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6960730);