#6567781

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

1278 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,33)(27,31,36,45)(28,38,40,48)(29,41,43,34)(32,42,47,39)(35,49,50,56)(37,53,52,46)(44,54,55,51);(1,3,5,4)(2,6,8,7)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,36)(26,31,33,45)(28,35,40,50)(29,37,43,52)(32,44,47,55)(34,46,41,53)(38,49,48,56)(39,51,42,54);(1,3,5,4)(2,7,8,6)(9,12,14,10)(11,15,19,17)(13,20,21,16)(18,22,24,23)(25,28,30,40)(26,32,33,47)(27,35,36,50)(29,42,43,39)(31,44,45,55)(34,48,41,38)(37,54,52,51)(46,56,53,49);(1,4,5,3)(2,6,8,7)(13,21)(16,20)(18,24)(22,23)(26,33)(31,45)(32,47)(34,41)(38,48)(44,55)(46,53)(49,56);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(27,36)(31,45)(35,50)(37,52)(44,55)(46,53)(49,56)(51,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,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, 6474977);