#255032

order	512
nilpotency class	3
isoclinism	isoclinic class of #246503
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	#59033	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1043 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)(3,6)(4,5)(7,8)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26)(27,35)(28,34)(29,33)(30,32)(31,38)(36,40)(37,39);(2,5)(6,8)(9,11,15,12)(10,16,19,21)(13,20,14,22)(17,18,24,23)(25,27,29,38)(26,31,33,35)(28,37,30,36)(32,39,34,40);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(2,5)(6,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,28,29,30)(26,32,33,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=eh,d^2=h,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=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 162228);