#144722

order	512
nilpotency class	4
isoclinism	isoclinic class of #142392
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#29742	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

499 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 64

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

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

Polycyclic presentation:

2^9|c^2=dg,e^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,d^c=di,e^c=egi,g^c=gi,h^c=hi,f^e=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51918);