C₂×#2252 #33723

order	256
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	Aut(G) order 262144
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

279 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4,6,8)(5,9,10,7)(12,16)(13,17)(14,20)(18,26)(19,24)(21,25)(22,23);(3,5,6,10)(4,7,8,9);(11,12,17,23,14,18,21,24,15,19,25,26,20,22,13,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26711);