#474588

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

797 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381784);