#417310

order	512
nilpotency class	3
isoclinism	isoclinic class of #416472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2864
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15248
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 324506);