C₅×#3392 #10616987

order	640
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₅
prime radical	C₅×#550	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×Q₈ ◁ G

404 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 192 normal subgroups are shown.

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

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(640, 19069);