#138937

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	C₂×#576	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

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

normal subgroups only

This group has 509 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=gh,e^2=ghi,f^2=gh,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, 46133);