#7167098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1750 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(22,27)(25,39)(28,41)(29,42)(30,43)(31,48)(34,44)(35,47)(37,40)(45,49)(46,52)(50,51);(7,13)(8,16)(10,17)(11,15)(21,22,25,27,26,30,39,43)(23,31,34,48,33,28,44,41)(24,35,37,47,38,42,40,29)(32,49,50,45,36,52,51,46);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,28)(24,36)(25,34)(26,33)(27,41)(29,46)(30,31)(32,38)(35,49)(37,51)(39,44)(40,50)(42,52)(43,48)(45,47);(7,13)(10,17)(14,19)(18,20)(21,24,26,38)(22,29,30,47)(23,32,33,36)(25,40,39,37)(27,42,43,35)(28,45,31,46)(34,51,44,50)(41,49,48,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,30)(23,33)(27,43)(29,47)(32,36)(34,44)(35,42)(50,51)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7074294);