#470565

order	512
nilpotency class	3
isoclinism	isoclinic class of #459490
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2733
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16689
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:

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

Polycyclic presentation:

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

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, 377761);