#138940

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

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

normal subgroups only

This group has 528 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,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.

Relationships

maximal subgroup of	#72280935
maximal normal subgroup of	#72280935

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46136);