#144761

order	512
nilpotency class	4
isoclinism	isoclinic class of #142404
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 366 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.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,3)(5,6,12,17)(7,16,20,29)(8,22,25,28)(9,18,11,15)(10,13,21,30)(14,19,32,23)(24,33,27,36)(26,31,35,34);(1,2)(3,4)(5,7,11,21,12,20,9,10)(6,13,18,29,17,30,15,16)(8,23,24,35,25,19,27,26)(14,28,31,33,32,22,34,36);(1,3,4,2)(5,8,9,24,12,25,11,27)(6,14,15,31,17,32,18,34)(7,19,10,26,20,23,21,35)(13,28,16,33,30,22,29,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,d^2=i,e^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fi,e^c=egi,g^c=gi,h^c=hi,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, 51957);