#6552251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,44,29,34,30,35)(27,36,45,47,38,32,39,51)(28,40,43,33,41,49,42,48)(37,52,55,50,53,56,54,46);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27,29,38)(26,32,34,36)(28,37,41,53)(30,39,31,45)(33,46,48,50)(35,47,44,51)(40,56,49,52)(42,54,43,55);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,29,41)(26,33,34,48)(27,37,38,53)(30,43,31,42)(32,46,36,50)(35,49,44,40)(39,55,45,54)(47,52,51,56);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(26,34)(32,36)(33,48)(35,44)(40,49)(46,50)(47,51)(52,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(26,34)(27,38)(33,48)(35,44)(37,53)(39,45)(40,49)(54,55)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=f,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459447);