#144850

order	512
nilpotency class	4
isoclinism	isoclinic class of #142220
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

549 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 549 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

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

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 20 points:

(2,3)(5,6,8,11)(7,12,9,10)(15,19);(1,2)(3,4)(5,7,8,9)(6,10,11,12)(14,17)(15,19)(18,20);(1,3,4,2)(6,10)(7,9)(11,12)(13,14)(15,20)(16,18)(17,19)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=dg,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi,h^c=hi,f^e=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52046);