#138079

order	512
nilpotency class	4
isoclinism	isoclinic class of #138078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9118
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

362 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,15,20,39)(4,22,26,43)(5,27,30,41)(6,12)(7,13)(8,14)(9,18)(10,25,38,31)(11,16,42,29)(17,49)(19,45,21,46)(23,57)(24,40)(28,37)(32,61,62,58)(33,47)(34,53)(35,54)(36,50,63,52)(44,55)(48,56)(51,64)(59,60);(1,3,16,47,6,21,5,18,8,20,29,53,7,19,30,54)(2,9,27,46,13,33,42,15,14,35,41,45,12,34,11,39)(4,23,48,52,31,51,28,61,26,55,59,50,25,17,24,58)(10,36,60,57,22,32,37,49,38,63,56,44,43,62,40,64);(1,4,7,25,8,26,6,31)(2,10,13,22,14,38,12,43)(3,17,21,23,20,51,19,55)(5,28,16,48,30,24,29,59)(9,32,34,36,35,62,33,63)(11,40,27,60,42,37,41,56)(15,44,46,49,39,57,45,64)(18,52,53,61,54,50,47,58)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=g,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,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, 45275);