C₂×#469 #3052

order	128
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

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

298 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1784);