#7597052

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	#61081	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1950 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)(4,5)(6,8)(9,10,16,22)(11,23,26,36)(12,19,29,34)(13,18,28,35)(14,21,15,20)(17,30,32,25)(24,37,38,31)(27,33,39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,25,16,26,15,30)(10,17,20,23,22,32,21,36)(12,27,28,38,29,39,13,24)(18,33,34,37,35,40,19,31);(2,5)(6,7)(9,12,15,13,16,29,14,28)(10,18,21,19,22,35,20,34)(11,24,25,27,26,38,30,39)(17,31,23,33,32,37,36,40);(1,3,4,8)(2,6,5,7)(9,12,14,28,16,29,15,13)(10,18,20,34,22,35,21,19)(11,24,30,39,26,38,25,27)(17,31,36,40,32,37,23,33);(9,13,15,29,16,28,14,12)(10,19,21,35,22,34,20,18)(11,24,30,39,26,38,25,27)(17,31,36,40,32,37,23,33)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=gi,e^2=fi,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, 7504248);