#6674312

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6581508);