#7565440

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519083
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	#24234	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fgi,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=cg,d^b=dh,g^b=gi,e^c=ei,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, 7472636);