#7562433

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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	#60646	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2100 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,13)(14,31)(15,20)(17,26)(18,29)(19,33)(21,38)(22,39)(23,37)(24,35)(25,28)(27,30)(32,34)(36,40);(2,5)(6,7)(9,11,15,24)(10,16,20,35)(12,25,27,39)(13,28,30,22)(14,32,33,23)(17,36,29,38)(18,21,26,40)(19,37,31,34);(9,12,14,26,15,27,33,18)(10,17,19,13,20,29,31,30)(11,21,23,39,24,40,32,25)(16,22,34,38,35,28,37,36);(9,13,14,29,15,30,33,17)(10,18,19,12,20,26,31,27)(11,22,23,38,24,28,32,36)(16,21,34,39,35,40,37,25);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

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, 7469629);