#452886

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16069
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

597 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360082);