#475718

order	512
nilpotency class	3
isoclinism	isoclinic class of #460330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

699 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:

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

Polycyclic presentation:

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

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, 382914);