#474345

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

553 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgh,c^2=ef,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 381541);