#144897

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

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

normal subgroups only

This group has 409 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)(3,6)(4,7)(5,8)(9,10,17,24)(11,25,31,47)(12,32,38,58)(13,23,16,20)(14,22,39,56)(15,21,41,54)(18,30,48,27)(19,43,53,35)(26,49,64,72)(28,57,42,45)(29,46,65,60)(33,44,68,70)(34,51,37,55)(36,50,40,52)(59,67,66,62)(61,69,63,71);(1,2)(3,7)(4,6)(5,8)(9,11,17,31)(10,18,24,48)(12,33,38,68)(13,30,16,27)(14,29,39,65)(15,42,41,28)(19,49,53,72)(20,47,23,25)(21,46,54,60)(22,57,56,45)(26,40,64,36)(32,66,58,59)(34,62,37,67)(35,63,43,61)(44,55,70,51)(50,69,52,71);(2,6)(3,4)(7,8)(9,12,13,34,17,38,16,37)(10,19,20,50,24,53,23,52)(11,26,27,61,31,64,30,63)(14,40,15,43,39,36,41,35)(18,44,25,59,48,70,47,66)(21,55,22,58,54,51,56,32)(28,33,65,67,42,68,29,62)(45,49,60,71,57,72,46,69)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=d,d^2=i,e^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=di,e^b=eh,f^b=fgi,g^b=gi,h^b=hi,e^c=egi,f^c=fg,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, 52093);