#244004

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,c^2=h,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151200);