C₂×#525 #3435

order	128
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center	C₂²×C₄	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵
socle	C₂³	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

940 conjugacy classes of subgroups; only the 452 normal subgroups are shown.

normal subgroups only

This group has 940 conjugacy classes of subgroups — too many to draw. Only its 452 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2167);