C₂×#3459 #62673

order	256
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶
socle	C₂³	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

4073 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55661);