#452976

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

929 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)(9,10,13,15,14,19,28,18)(11,20,27,32,23,30,22,16)(12,17,24,29,26,33,25,34)(21,36,38,31,39,35,37,40);(2,5)(3,7)(10,15,19,18)(12,24,26,25)(16,30,32,20)(21,37,39,38);(9,11)(10,16)(12,21)(13,27)(14,23)(15,20)(17,31)(18,30)(19,32)(22,28)(24,38)(25,37)(26,39)(29,35)(33,40)(34,36);(1,3,4,7)(2,6,5,8)(9,12,14,26)(10,17,19,33)(11,21,23,39)(13,24,28,25)(15,29,18,34)(16,31,32,40)(20,35,30,36)(22,37,27,38)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360172);