C₂×#2555 #34822

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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

2055 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);(4,5)(12,15)(16,18);(3,4,6,5)(7,8,9,10);(3,4,6,5)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18);(3,5,6,4)(11,13,14,17)(12,16,15,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27810);