#7565673

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519063
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

1926 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)(9,10,15,18)(11,16,22,32)(12,23,27,37)(13,14,29,25)(17,33,35,26)(19,34,36,40)(20,21,28,30)(24,31,39,38);(2,6)(7,8)(9,11,15,22)(10,16,18,32)(12,24,27,39)(13,28,29,20)(14,30,25,21)(17,34,35,40)(19,33,36,26)(23,31,37,38);(1,3,5,4)(2,7,6,8)(9,12,25,33,15,27,14,26)(10,17,29,37,18,35,13,23)(11,19,30,39,22,36,21,24)(16,31,28,40,32,38,20,34);(1,3,5,4)(2,7,6,8)(9,13,14,18,15,29,25,10)(11,20,21,32,22,28,30,16)(12,23,26,35,27,37,33,17)(19,34,24,38,36,40,39,31);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=fg,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,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7472869);