C₂×#2542 #34927

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

	subgroup	G/H
center, socle	C₂⁴	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵
prime radical	C₂³×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

2003 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);(3,4,5,6)(12,14)(13,17);(7,8,9,10)(11,12,15,14)(13,16,17,18);(3,4)(5,6)(7,8)(9,10)(11,13)(12,16)(14,18)(15,17);(11,14,15,12)(13,18,17,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27915);