C₅²×#463 #13103200

order	1600
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅² · Schur multiplier (H₂) = C₂²×C₅

Aliases

	subgroup	G/H
center, socle	C₂²×C₅²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₅²
prime radical	C₂×C₅²×D₈	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅² ◁ C₄²×C₅² ◁ G

496 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 800

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2,3,4,5)(12,14)(15,16)(17,19)(18,20)(21,22);(6,7,8,9,10)(11,12,13,14)(16,19)(20,21);(16,20)(18,22)(19,21)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1600, 5552);