C₂×#470 #3053

order	128
nilpotency class	3
isoclinism	isoclinic class of #468
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	C₂⁴	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

246 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1785);