#7111401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#28561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(26,31)(29,38)(32,33)(34,39)(35,37)(36,40);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,26,29,31,30,34,38,39)(27,32,35,33,28,36,37,40);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,27,30,28)(26,32,34,36)(29,35,38,37)(31,33,39,40);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,28,30,27)(26,33,34,40)(29,35,38,37)(31,32,39,36);(10,16)(11,18)(15,24)(17,19)(27,28)(32,36)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=hi,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=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, 7018597);