#144912

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

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

normal subgroups only

This group has 406 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) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=dg,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,e^b=eh,f^b=fg,g^b=gi,h^b=hi,d^c=di,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, 52108);