#453165

order	512
nilpotency class	3
isoclinism	isoclinic class of #427311
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16084
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 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)(3,9,15,25)(4,16,19,27)(5,22,23,29)(6,13)(7,11)(8,12)(10,20,28,21)(14,30)(17,24)(18,32)(26,31);(1,3,7,5,6,15,8,23)(2,9,11,22,13,25,12,29)(4,17,21,31,19,18,20,14)(10,26,27,32,28,30,16,24);(1,4,8,21,6,19,7,20)(2,10,11,16,13,28,12,27)(3,14,23,17,15,31,5,18)(9,24,22,30,25,32,29,26);(1,5,7,15,6,23,8,3)(2,9,11,22,13,25,12,29)(4,18,20,31,19,17,21,14)(10,24,16,30,28,32,27,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 360361);