#7111671

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1391 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)(25,26,31,27)(28,33,35,29)(30,36,32,34)(37,38,40,39);(2,5)(3,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,36,31,26,32,34)(28,29,37,38,35,33,40,39);(2,5)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,35)(26,33,27,29)(30,37,32,40)(34,39,36,38);(1,3,4,7)(2,6,5,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,28)(27,35)(30,39)(31,33)(32,38)(34,40)(36,37);(10,16)(11,18)(15,24)(17,19)(28,35)(29,33)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=hi,d^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,e^d=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, 7018867);