#7063115

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

1760 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)(9,10,14,20,15,19,13,29)(11,21,23,16,24,33,30,17)(12,18,27,38,28,36,31,37)(22,39,40,32,26,35,25,34);(1,2)(3,7)(4,6)(5,8)(9,11,15,24)(10,16,19,17)(12,25,28,40)(13,30,14,23)(18,35,36,39)(20,33,29,21)(22,31,26,27)(32,38,34,37);(1,3,4,8)(2,5,6,7)(9,11)(10,17)(12,26)(13,23)(14,30)(15,24)(16,19)(18,32)(20,33)(21,29)(22,28)(25,31)(27,40)(34,36)(35,37)(38,39);(3,8)(5,7)(9,12)(10,18)(11,22)(13,31)(14,27)(15,28)(16,32)(17,34)(19,36)(20,38)(21,39)(23,40)(24,26)(25,30)(29,37)(33,35);(2,6)(5,7)(10,19)(16,17)(18,36)(20,29)(21,33)(32,34)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,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, 6970311);