#416317

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323513);