#144755

order	512
nilpotency class	4
isoclinism	isoclinic class of #144754
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10396
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

341 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,e^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,e^c=eg,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51951);