#1304

order	128
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	Aut(G) = #72231982 (order 6144) · Inn(G) = #95 · Out(G) = C₂×GL(2,ℤ/4)
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#95
commutator	C₂³	C₄²
Frattini, prime radical, cosocle	C₂⁵	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

198 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^7|a^2=d,b^2=e,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 36);