#7119007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=ghi,e^2=g,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,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, 7026203);