#134477

order	512
nilpotency class	4
isoclinism	isoclinic class of #124648
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1804
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7,8,6)(4,5)(9,10)(11,23,26,17)(12,29,30,61)(13,34)(14,22)(15,21)(16,19)(18,31,44,63)(20,37)(24,52)(25,57,35,66)(27,42,28,43)(32,49,33,62)(36,46,65,45)(38,47)(39,50)(40,54)(41,59,48,60)(51,64)(53,68)(55,72)(56,71)(58,69)(67,70);(2,5)(6,7)(9,11,14,27,16,26,15,28)(10,17,22,43,19,23,21,42)(12,24,33,56,30,53,32,55)(13,35,37,60,39,25,38,59)(18,40,45,69,44,51,46,70)(20,48,50,66,47,41,34,57)(29,52,62,71,61,68,49,72)(31,64,36,67,63,54,65,58);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,24)(13,36)(14,33)(15,32)(16,30)(17,40)(19,44)(20,49)(21,46)(22,45)(23,51)(25,58)(26,53)(27,56)(28,55)(29,50)(31,38)(34,61)(35,67)(37,63)(39,65)(41,71)(42,70)(43,69)(47,62)(48,72)(52,66)(54,60)(57,68)(59,64)

Polycyclic presentation:

2^9|a^2=f,b^2=dg,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,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, 41673);