#137583

order	512
nilpotency class	4
isoclinism	isoclinic class of #137582
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9064
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#453
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

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

normal subgroups only

This group has 393 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,d^2=g,e^2=ghi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fgi,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, 44779);