#12556

order	256
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₈
commutator	C₈	C₂×C₄²
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂³	C₂×D₈
prime radical	C₂⁴	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#94341, #94343, #146331, …
maximal normal subgroup of	#94341, #94343, #146331, …
maximal quotient of	#126648, #126686, #126703, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5544);