#466143

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1069 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)(6,8)(10,16)(11,21)(13,14)(18,34)(19,20)(22,32)(23,37)(24,25)(26,38)(27,29)(30,36)(31,39);(1,2,4,5)(3,6,7,8)(9,10,13,16,15,20,14,19)(11,22,21,17,25,32,24,33)(12,18,29,30,28,36,27,34)(23,35,37,31,38,40,26,39);(3,7)(6,8)(9,11,15,25)(10,17,20,33)(12,26,28,37)(13,24,14,21)(16,22,19,32)(18,35,36,40)(23,29,38,27)(30,39,34,31);(1,3,4,7)(2,6,5,8)(9,12,15,28)(10,18,20,36)(11,23,25,38)(13,29,14,27)(16,30,19,34)(17,31,33,39)(21,37,24,26)(22,35,32,40)

Polycyclic presentation:

2^9|b^2=efg,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,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, 373339);