C₂×#2682 #52101

order	256
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1967 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);(3,4,5,6)(12,13)(16,17);(8,10)(11,12,15,13)(14,17,18,16);(3,4)(5,6)(7,8)(9,10)(11,13,15,12)(14,17,18,16);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45089);