#136513

order	512
nilpotency class	4
isoclinism	isoclinic class of #136511
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8921
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#455
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

338 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dgi,e^c=ehi,f^c=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43709);