#7194841

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1192 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,15,18,14)(12,20,22,16)(19,23,21,24)(25,26,31,28)(27,35,38,40)(29,34,30,33)(32,39,36,37);(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23)(25,27,31,38)(26,32,28,36)(29,37,30,39)(33,35,34,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,28,31,26)(27,36,38,32)(29,34,30,33)(35,37,40,39);(1,4,5,3)(2,6,7,8)(11,18)(14,15)(19,21)(23,24)(25,29)(26,33)(28,34)(30,31)(35,40)(37,39);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(26,28)(32,36)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7102037);