#7604484

order	512
nilpotency class	3
isoclinism	isoclinic class of #7563337
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	#532	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1626 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

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, 7511680);