#140977

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=ghi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,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, 48173);