#138938

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

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

normal subgroups only

This group has 504 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,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, 46134);