#474637

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=ef,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, 381833);