#473990

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=hi,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, 381186);