#452923

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2471
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16071
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

785 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=g,c^2=e,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, 360119);