C₂×#120 #521

order	64
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #72255861 (order 12288) · Inn(G) = C₂³ · Out(G) = C₂×#11714279
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

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

331 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 4 generators on 10 points:

(1,2);(4,6)(8,10);(7,8)(9,10);(3,4)(5,6)(7,8)(9,10)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 202);