C₂×#2008 #23410

order	256
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1059 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);(3,4)(5,13)(6,11)(7,10)(8,12)(9,14)(15,18)(16,17);(3,5)(4,9)(6,16)(7,14)(8,15)(10,18)(11,13)(12,17);(4,10)(6,7)(9,13)(11,12)(14,16)(17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16398);