#474775

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

857 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)(9,10,14,17,13,20,15,19)(11,16,23,29,22,32,27,31)(12,24,26,34,25,33,28,18)(21,35,37,30,36,40,38,39);(1,2,4,5)(3,6,7,8)(9,11,14,27,13,22,15,23)(10,16,17,31,20,32,19,29)(12,21,28,37,25,36,26,38)(18,30,33,40,34,39,24,35);(3,7)(6,8)(10,17,20,19)(12,25)(16,29,32,31)(18,33,34,24)(21,36)(26,28)(30,35,39,40)(37,38);(1,3,4,7)(2,6,5,8)(9,12,13,25)(10,18)(11,21,22,36)(14,28,15,26)(16,30)(17,33)(19,24)(20,34)(23,38,27,37)(29,35)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=g,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,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, 381971);