#146291

order	512
nilpotency class	4
isoclinism	isoclinic class of #146291
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂	#10690
commutator, Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(3,9,10,14)(4,12,13,6)(5,8)(7,11)(15,16);(2,6)(3,10)(7,14,15,9)(8,12);(1,3,4,11)(2,7,6,14)(5,10,13,16)(8,15,12,9)

Polycyclic presentation:

2^9|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,g^a=gi,c^b=cf,d^b=dh,e^b=eh,f^b=fg,e^c=eg,f^c=fh,h^c=hi,f^d=fi,f^e=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53487);