#7063109

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

1797 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)(9,10,14,21,15,19,32,20)(11,22,26,35,27,38,25,16)(12,28,31,36,13,18,30,17)(23,33,40,37,29,34,24,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,29)(13,23)(14,26)(15,27)(17,34)(18,37)(19,35)(20,38)(21,22)(24,31)(25,32)(28,39)(30,40)(33,36);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,23)(13,15)(14,30)(16,33)(18,21)(19,36)(20,28)(22,39)(24,26)(25,40)(27,29)(31,32)(34,35)(37,38);(3,8)(5,7)(9,13,15,12)(10,18,19,28)(11,24,27,40)(14,30,32,31)(16,34,35,33)(17,20,36,21)(22,39,38,37)(23,25,29,26);(2,6)(5,7)(10,19)(16,35)(17,36)(18,28)(20,21)(22,38)(33,34)(37,39)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,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, 6970305);