#7003081

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

1807 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)(11,21,25,33,26,38,24,16)(12,18,27,34,28,35,30,17)(22,31,39,36,40,32,23,37);(1,2)(3,7)(4,6)(5,8)(9,11,14,25,15,26,13,24)(10,16,29,38,19,33,20,21)(12,22,30,23,28,40,27,39)(17,31,18,36,34,32,35,37);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,22)(13,27)(14,30)(15,28)(16,31)(18,29)(19,34)(20,35)(21,37)(23,25)(24,39)(26,40)(32,33)(36,38);(3,8)(5,7)(9,12)(10,18)(11,23)(13,30)(14,27)(15,28)(16,32)(17,29)(19,35)(20,34)(21,37)(22,25)(24,40)(26,39)(31,33)(36,38);(2,6)(5,7)(10,19)(16,33)(17,34)(18,35)(20,29)(21,38)(31,32)(36,37)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 6910277);