C₂×#537 #3448

order	128
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

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

820 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 2180);