#291721

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59233	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

868 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)(3,7)(4,6)(5,8)(9,10,13,16)(11,17,18,24)(12,21,19,15)(14,22,23,20)(25,26)(27,32)(28,37)(29,33)(30,38)(31,34)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(11,18)(12,19)(15,21)(17,24)(25,27,31,36)(26,32,34,40)(28,29,35,30)(33,39,38,37);(9,11,19,22,13,18,12,20)(10,14,21,24,16,23,15,17)(25,28,30,27,31,35,29,36)(26,32,33,37,34,40,38,39);(2,6)(5,7)(11,18)(14,23)(17,24)(20,22)(25,29,31,30)(26,33,34,38)(27,35,36,28)(32,39,40,37)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198917);