C₂×#509 #3412

order	128
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

200 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2144);