#12168

order	256
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 37 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,d^b=df,e^b=eg,f^b=fh,d^c=dg,e^c=eh

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12168, #484636, #484638, …
maximal normal subgroup of	C₂×#12168, #484636, #484638, …
maximal quotient of	#123182, #123185, #123188, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5156);