#139429

order	512
nilpotency class	4
isoclinism	isoclinic class of #139428
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9428
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#3151	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

355 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=g,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=eg,f^c=fgh,h^c=hi,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46625);