#291467

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

718 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)(12,20,21,15)(14,22,23,19)(25,26,29,32)(27,30,34,39)(28,36,37,31)(33,40,35,38);(1,3,4,8)(2,5,6,7)(11,18)(12,21)(15,20)(17,24)(25,27,29,34)(26,30,32,39)(28,33,37,35)(31,38,36,40);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24)(27,33,34,35)(30,38,39,40);(2,6)(5,7)(11,18)(14,23)(17,24)(19,22)

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,c^2=f,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, 198663);