#6569195

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=fi,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^b=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476391);