#137562

order	512
nilpotency class	4
isoclinism	isoclinic class of #137561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9061
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3147	C₂²
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

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

normal subgroups only

This group has 436 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=g,e^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,g^b=gi,d^c=dgi,e^c=eghi,f^c=fgi,g^c=gi,h^c=hi

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, 44758);