#144924

order	512
nilpotency class	4
isoclinism	isoclinic class of #144895
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 68 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)(4,6)(7,8)(9,10,16,21)(11,17,26,48)(12,27,29,64)(13,33,14,38)(15,42,36,20)(18,31,50,30)(19,41,54,37)(22,49,57,71)(23,52,24,61)(25,55,43,47)(28,66,58,45)(32,53,44,65)(34,46,39,67)(35,51,40,56)(59,69,63,72)(60,68,62,70);(1,3)(2,5)(4,7)(6,8)(9,11,16,26)(10,17,21,48)(12,28,29,58)(13,34,14,39)(15,43,36,25)(18,49,50,71)(19,52,54,61)(20,55,42,47)(22,30,57,31)(23,41,24,37)(27,45,64,66)(32,62,44,60)(33,46,38,67)(35,63,40,59)(51,69,56,72)(53,68,65,70);(5,8)(9,12,15,44,16,29,36,32)(10,18,20,56,21,50,42,51)(11,22,25,63,26,57,43,59)(13,35,37,31,14,40,41,30)(17,45,47,70,48,66,55,68)(19,53,33,64,54,65,38,27)(23,60,34,28,24,62,39,58)(46,69,52,71,67,72,61,49)

Polycyclic presentation:

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

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, 52120);