C₂×#125 #527

order	64
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = #72226388 (order 8192) · Inn(G) = C₂³ · Out(G) = #72226368
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

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

113 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 208);