#7143625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993887
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

2159 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7050821);