#7188033

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1167 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=gi,d^2=i,e^2=h,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,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, 7095229);