#133276

order	512
nilpotency class	4
isoclinism	isoclinic class of #133275
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8524
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40472);