#7054931

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
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

2087 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)(9,10,16,20)(11,23,12,26)(13,19,31,36)(14,22,15,21)(17,28,18,25)(24,39,27,35)(29,34,40,33)(30,38,32,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,29)(14,28)(15,25)(18,20)(19,35)(21,26)(22,23)(24,30)(27,32)(31,40)(33,37)(34,38)(36,39);(1,3,4,8)(2,5,6,7)(9,12,16,11)(10,18,20,17)(13,24,31,27)(14,28,15,25)(19,33,36,34)(21,26,22,23)(29,32,40,30)(35,38,39,37);(3,8)(5,7)(9,13)(10,19)(11,24)(12,27)(14,32)(15,30)(16,31)(17,33)(18,34)(20,36)(21,38)(22,37)(23,39)(25,40)(26,35)(28,29);(2,6)(5,7)(10,20)(11,12)(19,36)(21,22)(24,27)(25,28)(29,40)(37,38)

Polycyclic presentation:

2^9|a^2=i,c^2=h,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=dh,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6962127);