C₂×#1705 #19592

order	256
nilpotency class	3
isoclinism	isoclinic class of #1705
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

	subgroup	G/H
center, socle	C₂³	#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

497 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2);(4,7)(5,9)(8,10)(12,15)(13,17)(16,18);(3,4,5,7,6,8,9,10);(11,12,13,15,14,16,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12580);