C₂×#126 #528

order	64
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #72255743 (order 6144) · Inn(G) = C₂³ · Out(G) = C₂×#2679⋊C₃
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

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

123 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);(5,9)(7,10)(12,15)(13,17);(3,4,6,8)(5,7,9,10)(11,12)(13,16)(14,15)(17,18);(3,5)(4,7)(6,9)(8,10)(11,13,14,17)(12,16,15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 209);