#131439

order	512
nilpotency class	4
isoclinism	isoclinic class of #131436
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

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

normal subgroups only

This group has 607 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,f^2=hi,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,e^c=ei,f^c=fg,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, 38635);