#474310

order	512
nilpotency class	3
isoclinism	isoclinic class of #453980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381506);