#291445

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

687 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)(9,10,14,18)(11,20,22,16)(12,21,23,24)(13,19,15,17)(25,26,29,32)(27,31,36,40)(28,33,30,37)(34,38,35,39);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,19)(13,21)(14,22)(15,24)(17,23)(18,20)(25,27)(26,31)(28,35)(29,36)(30,34)(32,40)(33,39)(37,38);(9,12,13,20,14,23,15,16)(10,11,17,24,18,22,19,21)(26,32)(27,34,36,35)(31,38,40,39)(33,37);(2,6)(5,7)(9,13,14,15)(10,17,18,19)(11,21,22,24)(12,16,23,20)(25,28,29,30)(26,33,32,37)(27,35,36,34)(31,39,40,38)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198641);