C₂×#123 #524

order	64
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #72225755 (order 4096) · Inn(G) = C₂³ · Out(G) = C₂×#13343
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

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

171 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 14 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 205);