#138939

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	#33522	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

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

normal subgroups only

This group has 517 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,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, 46135);