#131433

order	512
nilpotency class	4
isoclinism	isoclinic class of #131432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8404
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#3054	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

611 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,f^c=fgi,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38629);