#501405

order	512
nilpotency class	4
isoclinism	isoclinic class of #501404
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24553
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#32945	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=eh,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408601);