#7194822

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

1228 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)(26,32)(27,28)(29,30)(33,40)(34,37)(35,39);(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23)(25,26,31,35)(27,33,28,37)(29,32,30,39)(34,36,40,38);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,27,31,28)(26,33,35,37)(29,38,30,36)(32,34,39,40);(1,4,5,3)(2,6,7,8)(11,18)(14,15)(19,21)(23,24)(25,28,31,27)(26,34,35,40)(29,38,30,36)(32,33,39,37);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(25,29,31,30)(26,32,35,39)(27,36,28,38)(33,40,37,34)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=g,e^2=hi,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, 7102018);