#144757

order	512
nilpotency class	4
isoclinism	isoclinic class of #144756
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	#135
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 337 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,e^2=gi,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,f^b=fi,e^c=eg,f^c=fi,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, 51953);