#6571446

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	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

1307 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)(10,14)(15,17)(16,21)(22,23)(25,26,30,34,31,35,29,43)(27,36,39,47,38,48,44,32)(28,40,41,50,42,49,45,33)(37,46,54,56,52,55,53,51);(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)(26,32)(28,37)(29,39)(30,44)(31,38)(33,46)(34,48)(35,47)(36,43)(40,51)(41,53)(42,52)(45,54)(49,56)(50,55);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,31,42)(26,33,35,50)(27,37,38,52)(29,41,30,45)(32,46,47,55)(34,49,43,40)(36,51,48,56)(39,53,44,54);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(27,38)(32,47)(36,48)(37,52)(39,44)(46,55)(51,56)(53,54);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,31,30)(26,34,35,43)(27,39,38,44)(28,41,42,45)(32,48,47,36)(33,49,50,40)(37,53,52,54)(46,56,55,51)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=fg,d^2=fghi,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478642);