C₂×#2434 #34684

order	256
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

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

2643 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);(8,10)(11,12,15,18)(13,17,14,16);(4,6)(8,10)(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, 27672);