#452730

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

910 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)(9,10,15,19,14,20,13,16)(11,21,24,17,22,33,27,31)(12,18,25,34,26,32,28,29)(23,37,38,40,36,35,39,30);(1,3,5,8)(2,6,4,7)(10,16,20,19)(11,22)(12,25,26,28)(17,21,31,33)(18,32)(23,38,36,39)(24,27)(29,34);(1,4,5,2)(3,7,8,6)(9,11,14,22)(10,17,20,31)(12,23,26,36)(13,27,15,24)(16,21,19,33)(18,30,32,40)(25,38,28,39)(29,35,34,37);(3,8)(6,7)(9,12)(10,18)(11,23)(13,28)(14,26)(15,25)(16,29)(17,30)(19,34)(20,32)(21,35)(22,36)(24,38)(27,39)(31,40)(33,37)

Polycyclic presentation:

2^9|a^2=egh,b^2=efg,c^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=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, 359926);