C₂×#497 #3149

order	128
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
prime radical	C₂²×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

282 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);(4,6)(7,8,11,14)(9,15,16,22)(10,18,19,13)(12,20,21,17);(7,9,10,17,11,16,19,20)(8,12,13,15,14,21,18,22);(3,4,5,6)(9,16)(12,21)(15,22)(17,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1881);