#7054949

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6962145);