#7143214

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1964 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:

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

Polycyclic presentation:

2^9|d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7050410);