#137220

order	512
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9023
prime radical	C₂⁴	#137
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=g,e^2=i,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44416);