#467356

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16566
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

797 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,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, 374552);