#504522

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1753 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411718);