C₂²×#493 #60629

order	256
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

	subgroup	G/H
center, socle	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

2319 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53617);