#7188017

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, 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

1183 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)(26,32)(27,36)(29,30)(33,37)(34,40)(35,39);(9,11)(10,15)(12,14)(13,20)(16,18)(17,24)(19,22)(21,23)(25,26,29,39,31,35,30,32)(27,34,38,40,36,33,28,37);(1,3,4,8)(2,5,6,7)(9,12,14,11)(10,16,18,15)(13,20,22,19)(17,24,21,23)(25,27)(26,33)(28,30)(29,38)(31,36)(32,40)(34,35)(37,39);(11,12)(15,16)(19,20)(23,24)(25,28,31,38)(26,34,35,33)(27,29,36,30)(32,40,39,37);(9,13,14,22)(10,17,18,21)(11,19,12,20)(15,23,16,24)(25,29,31,30)(26,32,35,39)(27,28,36,38)(33,37,34,40)

Polycyclic presentation:

2^9|a^2=gi,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=df,e^b=eh,d^c=dh,e^c=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, 7095213);