#7566052

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	#61095	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 7473248);