#6564260

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	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 ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6471456);