C₂×#2546 #34928

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

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

1547 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27916);