#133270

order	512
nilpotency class	4
isoclinism	isoclinic class of #133267
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8522
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄×M₁₆	C₂³
prime radical	#3499	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 578 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,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,e^c=ei,f^c=fhi,f^d=fi,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40466);