#474083

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

722 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)(3,6)(4,5)(7,8)(9,10,15,20,16,22,14,21)(11,23,26,38,27,30,25,17)(12,28,31,18,33,39,32,40)(13,34,36,19,37,24,35,29);(2,5)(6,8)(9,11,16,27)(10,17)(12,29)(13,18,37,40)(14,25,15,26)(19,33)(20,23)(21,30)(22,38)(24,32)(28,36,39,35)(31,34);(3,7)(6,8)(9,12,15,31,16,33,14,32)(10,18,21,28,22,40,20,39)(11,24,25,19,27,34,26,29)(13,17,36,23,37,38,35,30);(1,3,4,7)(2,6,5,8)(9,13,14,35,16,37,15,36)(10,19,20,24,22,29,21,34)(11,18,25,28,27,40,26,39)(12,30,31,17,33,23,32,38)

Polycyclic presentation:

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

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, 381279);