#7054959

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

2022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|c^2=hi,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,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, 6962155);