C₂×#2363 #34135

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

2023 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27123);