#7054953

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
prime radical	#61076	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 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)(9,10,16,21)(11,22,25,27)(12,24,17,32)(13,18,31,37)(14,20,15,19)(23,39,35,34)(26,40,29,28)(30,38,33,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,21)(13,28)(14,32)(15,24)(16,25)(18,34)(19,27)(20,22)(23,30)(26,38)(29,36)(31,40)(33,35)(37,39);(1,3,4,8)(2,5,6,7)(9,12,16,17)(10,11,21,25)(13,29,31,26)(14,27,15,22)(18,35,37,23)(19,24,20,32)(28,36,40,38)(30,34,33,39);(3,8)(5,7)(9,13,16,31)(10,18,21,37)(11,23,25,35)(12,26,17,29)(14,33,15,30)(19,38,20,36)(22,39,27,34)(24,40,32,28);(2,6)(5,7)

Polycyclic presentation:

2^9|a^2=i,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=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, 6962149);