#445760

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=g,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, 352956);