#436728

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

862 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343924);