#410753

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15087
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

609 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=g,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 317949);