C₂×#137 #574

order	64
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) = #72216616 (order 1024) · Inn(G) = C₂×D₄ · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴
prime radical	C₂×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

129 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 255);