#140982

order	512
nilpotency class	4
isoclinism	isoclinic class of #140953
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 82 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)(4,8)(5,6)(9,10,15,21,16,22,14,20)(11,23,28,52,29,46,27,51)(12,30,33,45,34,49,37,18)(13,35,24,40,36,19,38,47)(17,39,42,58,43,53,48,26)(25,55,57,71,31,62,59,50)(32,65,63,67,66,61,54,44)(41,64,60,68,70,56,69,72);(1,3,7,2,5,8,4,6)(9,11,24,53)(10,17,40,52)(12,31,63,72)(13,26,15,27)(14,28,36,58)(16,29,38,39)(18,41,61,55)(19,46,20,48)(21,42,35,23)(22,43,47,51)(25,54,68,34)(30,60,67,50)(32,64,33,57)(37,59,66,56)(44,71,49,69)(45,70,65,62);(2,6)(3,8)(9,12,14,33,16,34,15,37)(10,18,21,49,22,45,20,30)(11,25,28,59,29,31,27,57)(13,32,24,54,36,66,38,63)(17,41,42,69,43,70,48,60)(19,44,40,65,35,67,47,61)(23,50,51,55,46,71,52,62)(26,56,53,72,58,64,39,68)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48178);