#7143632

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993899
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2198 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)(9,10,14,18)(11,20,24,38)(12,17,27,35)(13,28,29,19)(15,30,33,23)(16,25,34,26)(21,31,40,32)(22,37,39,36);(2,5)(3,7)(9,11,13,23,14,24,29,30)(10,15,19,20,18,33,28,38)(12,21,25,39,27,40,26,22)(16,31,35,37,34,32,17,36);(1,2)(3,6)(4,5)(7,8)(9,12)(10,16)(11,21)(13,25)(14,27)(15,31)(17,28)(18,34)(19,35)(20,37)(22,30)(23,39)(24,40)(26,29)(32,33)(36,38);(2,5)(6,8)(9,12)(10,17)(11,22)(13,26)(14,27)(15,32)(16,28)(18,35)(19,34)(20,37)(21,30)(23,40)(24,39)(25,29)(31,33)(36,38);(1,3,4,7)(2,6,5,8)(10,18)(15,33)(16,34)(17,35)(19,28)(20,38)(31,32)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=h,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,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, 7050828);