#8793

order	256
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#382
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(2,3)(4,5)(7,8)(9,10)(11,19,23,15)(12,24,26,39)(13,18)(14,17)(16,25,30,34)(20,27)(21,35)(22,32,33,29)(28,36)(31,40)(37,38);(1,2,4,3,6,8,5,7)(9,11,20,33,14,23,13,22)(10,15,27,32,17,19,18,29)(12,21,34,40,26,37,25,36)(16,28,39,35,30,31,24,38);(1,3,5,2,6,7,4,8)(9,12,14,26)(10,16)(11,21,23,37)(13,25,20,34)(15,28)(17,30)(18,24)(19,31)(22,36,33,40)(27,39)(29,38)(32,35)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#134987, #134994, #134995, …
maximal normal subgroup of	#134987, #134994, #134995, …
maximal quotient of	#99279, #99280, #99286, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1781);