#476360

order	512
nilpotency class	3
isoclinism	isoclinic class of #476109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1011 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,h^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=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,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, 383556);