#141502

order	512
nilpotency class	4
isoclinism	isoclinic class of #141499
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9785
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

317 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,15)(4,22,27,58)(5,29,33,52)(6,34)(7,13)(8,14)(9,18,39,30)(10,26,42,40)(11,16,44,50)(12,36)(17,51,53,32)(19,48)(20,46)(21,47)(23,55,61,49)(24,45)(25,56)(28,59,64,54)(31,60)(35,63)(37,62)(38,43)(41,57);(1,3,16,10,7,37,33,18,8,21,50,42,38,20,5,30)(2,9,29,62,43,40,11,15,14,39,52,46,13,26,44,47)(4,23,48,34,28,17,25,41,27,61,63,45,64,53,31,12)(6,35,49,58,36,56,51,59,24,19,55,22,57,60,32,54);(1,4,24,40,7,28,57,39,8,27,6,26,38,64,36,9)(2,10,41,58,43,18,45,59,14,42,12,22,13,30,34,54)(3,17,35,29,37,61,56,11,21,53,19,52,20,23,60,44)(5,31,51,62,16,48,55,15,33,25,32,46,50,63,49,47)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48698);