#7054921

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
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

2063 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)(11,23,26,38)(12,28,13,19)(14,22,15,21)(17,32,35,27)(18,31,36,30)(24,33,29,34)(25,39,40,37);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,29)(13,24)(14,32)(15,27)(16,26)(18,34)(19,37)(20,35)(21,38)(22,23)(25,31)(28,39)(30,40)(33,36);(1,3,4,8)(2,5,6,7)(9,12)(10,18)(11,24)(13,16)(14,30)(15,31)(17,33)(19,22)(20,36)(21,28)(23,39)(25,32)(26,29)(27,40)(34,35)(37,38);(3,8)(5,7)(9,13,16,12)(10,19,20,28)(11,25,26,40)(14,30,15,31)(17,34,35,33)(18,21,36,22)(23,39,38,37)(24,27,29,32);(2,6)(5,7)(10,20)(11,26)(18,36)(19,28)(21,22)(24,29)(25,40)(27,32)

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=dh,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, 6962117);