#134491

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

391 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41687);