#7597031

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563319
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	#61084	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2002 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)(11,21)(12,18)(13,30)(14,32)(15,20)(16,24)(17,28)(19,31)(22,33)(23,40)(25,35)(26,34)(27,38)(29,37)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,24,15,26,31,25)(10,16,19,21,20,35,32,34)(12,27,28,22,29,39,13,23)(17,36,18,33,30,38,37,40);(2,5)(6,7)(9,12,14,28,15,29,31,13)(10,17,19,18,20,30,32,37)(11,22,25,27,26,23,24,39)(16,33,34,36,35,40,21,38);(1,3,4,8)(2,6,5,7)(9,12,14,28,15,29,31,13)(10,17,19,18,20,30,32,37)(11,22,25,27,26,23,24,39)(16,33,34,36,35,40,21,38);(9,13,31,29,15,28,14,12)(10,18,32,17,20,37,19,30)(11,23,25,39,26,22,24,27)(16,33,34,36,35,40,21,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504227);