#7125415

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1740 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)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,26,30)(23,31,34,41)(24,28,36,43)(25,38,39,29)(27,33,42,35)(32,45,48,51)(37,50,49,44)(40,46,52,47);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,35,26,34,39,33)(22,27,29,31,30,42,38,41)(24,32,37,47,36,48,49,46)(28,40,44,45,43,52,50,51);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,26,36)(22,28,30,43)(23,32,34,48)(25,37,39,49)(27,40,42,52)(29,44,38,50)(31,45,41,51)(33,46,35,47);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(23,33)(25,39)(27,41)(29,38)(31,42)(32,46)(34,35)(37,49)(40,51)(44,50)(45,52)(47,48);(6,12)(10,13)(11,20)(18,19)(23,34)(24,36)(27,42)(28,43)(31,41)(33,35)(37,49)(44,50)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 7032611);