#446277

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

890 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=efgi,d^2=ef,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,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353473);