C₂×#460 #3032

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	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

292 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);(4,6)(8,11)(9,13)(12,14);(3,4,5,6);(7,8,10,12)(9,14,13,11)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1764);