#478277

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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:

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

Polycyclic presentation:

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

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, 385473);