#138098

order	512
nilpotency class	4
isoclinism	isoclinic class of #138098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45294);