#144880

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.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#30247	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

496 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52076);