#467491

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

827 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)(3,6)(4,5)(7,8)(9,10,14,19,15,20,13,18)(11,16,24,32,25,33,23,21)(12,26,30,17,28,34,29,27)(22,36,39,40,37,35,38,31);(9,11,14,24,15,25,13,23)(10,16,19,32,20,33,18,21)(12,22,30,39,28,37,29,38)(17,31,34,36,27,40,26,35);(2,5)(6,8)(9,10,15,20)(11,21,25,32)(12,27,28,17)(13,19,14,18)(16,24,33,23)(22,36,37,35)(26,30,34,29)(31,38,40,39);(1,3,4,7)(2,6,5,8)(9,12,13,29,15,28,14,30)(10,17,18,26,20,27,19,34)(11,22,23,38,25,37,24,39)(16,31,21,35,33,40,32,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 374687);