#469961

order	512
nilpotency class	3
isoclinism	isoclinic class of #469927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2605
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16671
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

589 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)(3,13,14,25,16,23,17,8)(4,18,20,27,22,28,21,9)(15,24,31,32,19,26,30,29);(3,14,16,17)(8,23,25,13)(15,30,19,31)(24,32,26,29);(1,3,5,17,7,16,6,14)(2,8,10,23,12,25,11,13)(4,19,20,31,22,15,21,30)(9,26,18,32,27,24,28,29);(1,4,5,21,7,22,6,20)(2,9,11,18,12,27,10,28)(3,15,17,31,16,19,14,30)(8,24,13,29,25,26,23,32)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377157);