#454086

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2168 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)(3,4)(5,6)(7,8)(9,10,14,20)(11,23,26,17)(12,28,13,31)(15,22,16,21)(18,32,19,30)(24,35,40,37)(25,34,29,39)(27,36,33,38);(6,8)(9,11)(10,17)(12,29)(13,25)(14,26)(15,33)(16,27)(18,37)(19,35)(20,23)(21,38)(22,36)(24,30)(28,34)(31,39)(32,40);(2,4)(5,6)(7,8)(9,12)(10,18)(11,24)(13,14)(15,32)(16,30)(17,34)(19,20)(21,31)(22,28)(23,39)(25,27)(26,40)(29,33)(35,36)(37,38);(6,8)(9,13,14,12)(10,19,20,18)(11,25,26,29)(15,32,16,30)(17,35,23,37)(21,31,22,28)(24,33,40,27)(34,38,39,36)

Polycyclic presentation:

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

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, 361282);