#138941

order	512
nilpotency class	4
isoclinism	isoclinic class of #138936
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9259
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#33524	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 507 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=ghi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fhi,g^b=gi,d^c=dgh,e^c=egh,f^c=fgh,h^c=hi,f^d=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, 46137);