#7167122

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1759 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)(22,28)(25,26)(29,41)(30,36)(31,32)(34,39)(37,40)(42,47)(43,46)(44,45)(48,49)(51,52);(7,13)(8,16)(10,17)(11,15)(21,22,25,31,27,32,26,28)(23,29,39,46,35,43,34,41)(24,36,40,45,38,44,37,30)(33,47,48,52,50,51,49,42);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,29)(24,33)(25,39)(26,34)(27,35)(28,41)(30,42)(31,46)(32,43)(36,47)(37,49)(38,50)(40,48)(44,51)(45,52);(7,13)(10,17)(14,19)(18,20)(21,24,27,38)(22,30,32,45)(23,33,35,50)(25,37,26,40)(28,36,31,44)(29,42,43,52)(34,48,39,49)(41,47,46,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,27,26)(22,31,32,28)(23,34,35,39)(24,37,38,40)(29,41,43,46)(30,44,45,36)(33,48,50,49)(42,47,52,51)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7074318);