#352793

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=fi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,d^b=dhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259989);