#474082

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 381278);