#144677

order	512
nilpotency class	4
isoclinism	isoclinic class of #144674
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,15,20)(11,21,24,43)(12,26,29,17)(13,31,34,18)(14,19,36,46)(16,32,38,23)(22,47,51,70)(25,48,35,66)(27,56,58,68)(28,41,60,54)(30,42,49,55)(33,62,65,44)(37,63,67,50)(39,64,45,52)(40,57,53,59)(61,72,71,69);(9,11,15,24)(10,16,20,38)(12,27,29,58)(13,32,34,23)(14,35,36,25)(17,40,26,53)(18,43,31,21)(19,45,46,39)(22,49,51,30)(28,59,60,57)(33,52,65,64)(37,55,67,42)(41,56,54,68)(44,48,62,66)(47,69,70,72)(50,61,63,71);(2,5)(3,8)(6,7)(9,12,13,28,15,29,34,60)(10,17,18,41,20,26,31,54)(11,22,23,50,24,51,32,63)(14,30,33,61,36,49,65,71)(16,37,21,47,38,67,43,70)(19,42,44,69,46,55,62,72)(25,27,52,57,35,58,64,59)(39,40,48,68,45,53,66,56)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51873);