#144144

order	512
nilpotency class	4
isoclinism	isoclinic class of #144131
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fghi,h^b=hi,d^c=dg,e^c=eh,f^c=fghi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51340);