#140975

order	512
nilpotency class	4
isoclinism	isoclinic class of #140973
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ 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)(3,5,7,8)(4,6)(9,10)(11,22,27,48)(12,28,32,60)(13,34)(14,20)(15,39)(16,21)(17,25,43,35)(18,31,47,33)(19,36)(23,49)(24,55)(26,51,38,52)(29,61)(30,65,67,46)(37,40)(41,63)(42,53,44,69)(45,56)(50,64)(54,70,62,59)(57,72)(58,66)(68,71);(1,3,4,7)(2,5)(6,8)(9,11,23,53,14,38,37,25,16,27,36,69,15,26,13,35)(10,17,40,52,39,42,49,22,21,43,34,51,20,44,19,48)(12,29,62,56,31,24,30,66,32,64,54,68,33,57,67,63)(18,45,59,50,60,41,46,55,47,71,70,61,28,58,65,72);(2,6)(3,7)(9,12,16,32)(10,18)(11,24)(13,30,37,67)(14,31,15,33)(17,41,43,58)(19,46)(20,28)(21,47)(22,50,48,61)(23,54,36,62)(25,56)(26,29)(27,57)(34,59)(35,68)(38,64)(39,60)(40,70)(42,71,44,45)(49,65)(51,55,52,72)(53,63)(66,69)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=egi,d^2=ghi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=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, 48171);