#137217

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#9023
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

327 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,12,7,13,29,41,8,14,5,11,26,31,27,39)(3,15,17,34,18,38,25,33,19,44,50,42,30,40,28,9)(4,20,23,35,24,52,57,61,22,10,21,36,54,37,55,63)(16,45,46,64,48,62,53,32,47,43,49,59,58,60,56,51);(1,3,7,30,8,19,26,18)(2,9,31,34,14,33,13,42)(4,16,24,58,22,47,54,48)(5,25,27,17,6,28,29,50)(10,32,52,59,20,51,37,64)(11,38,41,44,12,40,39,15)(21,53,55,46,23,56,57,49)(35,60,63,45,36,62,61,43);(1,4,8,22)(2,10)(3,16,19,47)(5,23,6,21)(7,24,26,54)(9,32)(11,35)(12,36)(13,37)(14,20)(15,43)(17,49,50,46)(18,48,30,58)(25,56,28,53)(27,57,29,55)(31,52)(33,51)(34,59)(38,60)(39,61)(40,62)(41,63)(42,64)(44,45)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=h,d^2=g,f^2=g,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44413);