#291722

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	#59228	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

827 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)(9,10,14,19)(11,21,17,16)(12,23,24,22)(13,20,15,18)(25,26)(27,35)(28,31)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11,14,17)(10,16,19,21)(12,20,24,18)(13,22,15,23)(26,31)(27,36)(29,30)(32,35)(37,38)(39,40);(9,12,13,16,14,24,15,21)(10,17,18,22,19,11,20,23)(25,27,29,38,28,37,30,36)(26,32,33,35,31,39,34,40);(2,6)(5,7)(9,13,14,15)(10,18,19,20)(11,22,17,23)(12,21,24,16)(26,31)(27,37)(33,34)(36,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=fhi,d^2=hi,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, 198918);