#6443964

order	512
nilpotency class	3
isoclinism	isoclinic class of #6439025
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	#23400	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351160);