#7565436

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	#3500	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1703 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)(4,8)(5,6)(9,10)(11,16)(12,23)(13,27)(14,28)(15,18)(17,26)(19,33)(20,21)(22,32)(24,30)(25,38)(29,31)(34,35)(36,37)(39,40);(2,6)(7,8)(9,11,15,22)(10,16,18,32)(12,24,25,39)(13,20,28,31)(14,29,27,21)(17,33,34,36)(19,35,37,26)(23,30,38,40);(1,3,5,4)(2,7,6,8)(9,12,14,26,15,25,27,35)(10,17,13,23,18,34,28,38)(11,19,21,24,22,37,29,39)(16,30,31,33,32,40,20,36);(1,3,5,4)(2,7,6,8)(9,13,27,10,15,28,14,18)(11,20,29,32,22,31,21,16)(12,23,35,17,25,38,26,34)(19,36,39,40,37,33,24,30);(1,4,5,3)(2,7,6,8)(12,25)(17,34)(19,37)(23,38)(24,39)(26,35)(30,40)(33,36)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,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, 7472632);