#479770

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

632 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,h^c=hi,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, 386966);