#453670

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

717 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:

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

Polycyclic presentation:

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

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, 360866);