#7560691

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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
prime radical	#61168	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(3,6)(4,5)(7,8)(9,10)(11,20)(12,23)(13,26)(14,18)(15,31)(16,22)(17,29)(19,30)(21,36)(24,35)(25,34)(27,37)(28,40)(32,39)(33,38);(2,5)(6,8)(9,11,14,22)(10,16,18,20)(12,15,25,30)(13,27,28,21)(17,32,33,35)(19,34,31,23)(24,38,39,29)(26,36,40,37);(1,3,4,7)(2,6,5,8)(9,12,15,22,14,25,30,11)(10,16,19,23,18,20,31,34)(13,24,29,27,28,39,38,21)(17,32,26,37,33,35,40,36);(9,13,15,29,14,28,30,38)(10,17,19,26,18,33,31,40)(11,21,12,24,22,27,25,39)(16,32,23,37,20,35,34,36);(2,5)(6,8)(10,18)(16,20)(17,33)(19,31)(23,34)(26,40)(32,35)(36,37)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=g,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467887);