C₂×#2104 #23846

order	256
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	Aut(G) order 57344
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²	#401
commutator, Frattini, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

411 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)(5,11,13,8)(6,9,15,18)(7,10)(12,17)(14,16);(3,5)(4,8)(6,14,15,12)(7,13)(9,17,18,16)(10,11);(3,6)(4,9,10,18)(5,12)(7,15)(8,16,11,17)(13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16834);