C₂×#398 #2391

order	128
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

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

346 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1123);