#6569221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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

1223 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=f,d^2=fgi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476417);