#7565681

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519075
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

1927 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)(12,23,26,40)(13,14,28,30)(17,33,35,25)(19,36,38,34)(20,21,27,29)(24,39,37,31);(2,6)(7,8)(9,11,15,22)(10,16,18,32)(12,24,26,37)(13,27,28,20)(14,29,30,21)(17,34,35,36)(19,25,38,33)(23,39,40,31);(1,3,5,4)(2,7,6,8)(9,12,14,25,15,26,30,33)(10,17,13,23,18,35,28,40)(11,19,21,37,22,38,29,24)(16,31,20,36,32,39,27,34);(1,3,5,4)(2,7,6,8)(9,13,14,18,15,28,30,10)(11,20,21,32,22,27,29,16)(12,23,25,35,26,40,33,17)(19,36,37,39,38,34,24,31);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,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=cgi,d^b=dh,g^b=gi,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, 7472877);