#133264

order	512
nilpotency class	4
isoclinism	isoclinic class of #133263
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	#60831	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 581 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)(3,14,17,45,18,8,31,34,19,33,52,41,16,35,27,56)(4,20,22,53,23,39,55,62,24,37,28,58,21,9,36,38)(15,46,48,64,47,44,57,63,49,32,51,59,50,60,54,61);(1,3)(2,8)(4,15)(5,16)(6,27)(7,19)(9,32)(10,33)(11,41)(12,14)(13,35)(17,30)(18,26)(20,44)(21,47)(22,54)(23,50)(24,49)(25,52)(28,57)(29,31)(34,42)(36,48)(37,60)(38,61)(39,46)(40,56)(43,45)(51,55)(53,59)(58,64)(62,63);(1,4,7,24)(2,9,13,39)(3,15,19,49)(5,23,26,21)(6,28,29,22)(8,32,35,46)(10,37,12,20)(11,38,43,62)(14,44,33,60)(16,50,18,47)(17,51,52,48)(25,36,30,55)(27,57,31,54)(34,59,56,64)(40,58,42,53)(41,61,45,63)

Polycyclic presentation:

2^9|a^2=f,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,f^b=fghi,g^b=gi,e^c=ei,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, 40460);