#7054929

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

2089 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)(10,16)(11,20)(13,14)(18,30)(19,26)(21,35)(22,28)(23,34)(24,27)(29,33)(36,39)(37,38);(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,23)(13,26)(14,16)(15,19)(18,24)(20,33)(21,36)(22,32)(25,30)(27,34)(28,29)(31,38)(35,40)(37,39);(1,3,4,8)(2,5,6,7)(9,11,15,22)(10,17,19,32)(12,21,25,37)(13,20,14,28)(16,29,26,33)(18,31,34,40)(23,36,30,39)(24,38,27,35);(3,8)(5,7)(9,12,15,25)(10,18,19,34)(11,21,22,37)(13,27,14,24)(16,30,26,23)(17,31,32,40)(20,35,28,38)(29,39,33,36);(2,6)(5,7)(11,22)(17,32)(20,28)(21,37)(29,33)(31,40)(35,38)(36,39)

Polycyclic presentation:

2^9|c^2=hi,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=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, 6962125);