#416741

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 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)(7,17,20,12)(8,22)(9,16)(10,15)(11,14)(13,25)(18,26,21,28)(19,33,35,36)(23,30)(24,29)(27,34,31,32);(1,2)(3,4)(5,7,10,21,11,20,9,18)(6,12,15,26,14,17,16,28)(8,19,25,34,23,35,24,32)(13,27,30,33,29,31,22,36);(2,4)(7,18,20,21)(8,23)(12,26,17,28)(13,29)(19,34,35,32)(22,30)(24,25)(27,36,31,33);(5,8,10,25,11,23,9,24)(6,13,16,22,14,29,15,30)(7,19,21,34,20,35,18,32)(12,27,28,36,17,31,26,33)

Polycyclic presentation:

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

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, 323937);