#475049

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	#23386	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1097 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)(11,19,21,15)(12,23,24,38)(13,18)(14,17)(16,25,29,39)(20,33)(22,32,36,28)(26,30)(27,34)(31,40)(35,37);(2,5)(6,8)(9,11,13,22,14,21,26,36)(10,15,18,28,17,19,30,32)(12,20,25,34,24,35,39,40)(16,27,23,37,29,31,38,33);(1,3)(2,6)(4,7)(5,8)(15,19)(16,29)(23,38)(28,32);(3,7)(6,8)(9,12)(10,16,17,29)(11,20)(13,25)(14,24)(15,27,19,31)(18,23,30,38)(21,35)(22,34)(26,39)(28,37,32,33)(36,40)

Polycyclic presentation:

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

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, 382245);