C₂²×#498 #60652

order	256
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1199 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2);(3,4);(5,6,8,10)(7,11,12,9)(14,17)(15,19)(18,20);(5,7,8,12)(6,9,10,11);(13,14,15,17,16,18,19,20)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53640);