#7180367

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	#24662	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=f,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7087563);