#7143626

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

2104 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)(11,21,25,40)(12,26,30,36)(13,18,33,23)(14,34,22,19)(16,35,27,24)(17,28,37,29)(31,39,32,38);(2,5)(3,7)(9,11,22,35,15,25,14,24)(10,16,34,40,20,27,19,21)(12,23,29,38,30,18,28,39)(13,26,32,37,33,36,31,17);(1,2)(3,6)(4,5)(7,8)(9,12,15,30)(10,17,20,37)(11,23,25,18)(13,27,33,16)(14,28,22,29)(19,36,34,26)(21,31,40,32)(24,39,35,38);(2,5)(6,8)(9,13)(10,18)(11,17)(12,27)(14,32)(15,33)(16,30)(19,39)(20,23)(21,28)(22,31)(24,26)(25,37)(29,40)(34,38)(35,36);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,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, 7050822);