#479793

order	512
nilpotency class	3
isoclinism	isoclinic class of #478348
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2754
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17252
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

501 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,c^2=f,d^2=e,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,c^b=cg,d^b=def,d^c=de,e^c=ei,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386989);