#7565065

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	#3499	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1695 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)(9,10,15,19)(11,16,23,33)(12,24,28,26)(13,29,18,14)(17,34,36,27)(20,37,39,35)(21,31,30,22)(25,40,38,32);(2,6)(7,8)(9,11,15,23)(10,16,19,33)(12,25,28,38)(13,30,18,21)(14,31,29,22)(17,35,36,37)(20,27,39,34)(24,40,26,32);(1,3,5,4)(2,7,6,8)(9,12,14,27,15,28,29,34)(10,17,18,24,19,36,13,26)(11,20,22,38,23,39,31,25)(16,32,30,37,33,40,21,35);(1,3,5,4)(2,7,6,8)(9,13,14,10,15,18,29,19)(11,21,22,16,23,30,31,33)(12,26,27,17,28,24,34,36)(20,35,38,32,39,37,25,40);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=i,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, 7472261);