#1438

order	128
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

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

584 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^7|a^2=g,b^a=bd,c^a=ce,c^b=cf

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 170);