#507797

order	512
nilpotency class	4
isoclinism	isoclinic class of #500599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 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)(3,7)(4,5)(6,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,27)(22,37)(23,36)(25,32)(28,40)(29,31)(30,33)(34,38)(35,39);(2,5)(6,7)(9,11,13,24,15,23,14,25)(10,16,19,21,20,32,18,36)(12,27,28,22,30,39,29,38)(17,33,34,40,35,26,37,31);(9,12,15,30)(10,17,20,35)(11,22,23,38)(13,29,14,28)(16,31,32,40)(18,34,19,37)(21,26,36,33)(24,27,25,39);(1,3,4,8)(2,6,5,7)(10,18,20,19)(11,23)(16,21,32,36)(17,34,35,37)(22,38)(24,25)(26,40,33,31)(27,39)

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=eg,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^b=fi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,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, 414993);