#471073

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

641 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=i,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378269);