#6569180

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1237 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)(25,26,30,35)(27,36,38,50)(28,41,43,48)(29,33,44,34)(31,39,46,40)(32,47,49,42)(37,52,53,45)(51,54,56,55);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,40,30,38,44,39)(26,31,34,36,35,46,33,50)(28,37,42,55,43,53,47,54)(32,45,41,51,49,52,48,56);(1,3,5,4)(2,7,6,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,42,30,43,44,47)(26,32,34,41,35,49,33,48)(27,37,40,55,38,53,39,54)(31,45,36,51,46,52,50,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(31,46)(36,50)(37,53)(39,40)(45,52)(51,56)(54,55);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,30,44)(26,33,35,34)(27,39,38,40)(28,42,43,47)(31,36,46,50)(32,48,49,41)(37,54,53,55)(45,51,52,56)

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,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^b=ei,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, 6476376);