#2104

order	128
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	Aut(G) = #72237983 (order 3584) · Inn(G) = #401 · Out(G) = AGL(1,𝔽₈)
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂	#401
commutator, Frattini, prime radical, cosocle	C₂⁴	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

89 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2)(3,9,11,6)(4,7,13,16)(5,8)(10,15)(12,14);(1,3)(2,6)(4,12,13,10)(5,11)(7,15,16,14)(8,9);(1,4)(2,7,8,16)(3,10)(5,13)(6,14,9,15)(11,12)

Polycyclic presentation:

2^7|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=cf,f^b=fg,e^c=eg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 836);