C₂×#462 #3033

order	128
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(4,6)(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16);(3,4,5,6);(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1765);