#7063105

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

1785 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)(9,10,15,20,16,19,14,30)(11,21,23,17,12,24,26,33)(13,18,28,38,29,36,31,37)(22,39,40,32,25,34,27,35);(1,2)(3,7)(4,6)(5,8)(9,11)(10,17)(12,16)(13,27)(14,26)(15,23)(18,34)(19,33)(20,24)(21,30)(22,28)(25,31)(29,40)(32,37)(35,38)(36,39);(1,3,4,8)(2,5,6,7)(9,12,16,11)(10,17,19,33)(13,22,29,25)(14,26,15,23)(18,35,36,32)(20,21,30,24)(27,31,40,28)(34,37,39,38);(3,8)(5,7)(9,13,16,29)(10,18,19,36)(11,22,12,25)(14,31,15,28)(17,32,33,35)(20,38,30,37)(21,39,24,34)(23,40,26,27);(2,6)(5,7)(10,19)(17,33)(18,36)(20,30)(21,24)(32,35)(34,39)(37,38)

Polycyclic presentation:

2^9|a^2=g,c^2=h,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, 6970301);