C₄×#125 #2305

order	128
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

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

242 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,3,4);(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20);(6,9)(10,12)(13,15,16,19)(14,17,18,20);(5,7,8,11)(6,10,9,12)(14,18)(17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1037);