#452410

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	#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

792 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)(9,10,16,20,15,21,14,31)(11,22,33,35,26,40,25,17)(12,18,27,24,28,37,34,36)(13,19,32,23,30,38,29,39);(1,3)(2,5)(4,7)(6,8)(9,11,14,25,15,26,16,33)(10,17,20,22,21,35,31,40)(12,23,27,38,28,39,34,19)(13,18,29,36,30,37,32,24);(3,7)(5,8)(9,12,16,34,15,28,14,27)(10,18,20,36,21,37,31,24)(11,23,33,19,26,39,25,38)(13,17,29,22,30,35,32,40);(9,13,16,29,15,30,14,32)(10,19,20,39,21,38,31,23)(11,24,33,18,26,36,25,37)(12,17,34,22,28,35,27,40)

Polycyclic presentation:

2^9|a^2=egh,b^2=e,c^2=ef,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, 359606);