C₂×#487 #3129

order	128
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

212 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1861);