C₅×#10528 #11933248

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10528

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

342 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

(1,2,3,4,5)(6,7)(8,11)(9,12)(10,13)(14,15,20,24)(16,25,29,40)(17,30,33,22)(18,35,19,23)(21,28,37,27)(26,41,43,38)(31,36,42,45)(32,39,34,44);(6,7)(8,12)(9,11)(10,13)(14,16,20,29)(15,21,24,37)(17,31,33,42)(18,28,19,27)(22,38,30,41)(23,40,35,25)(26,32,43,34)(36,44,45,39);(7,11)(8,9)(12,13)(14,17,18,32,20,33,19,34)(15,22,23,39,24,30,35,44)(16,26,27,42,29,43,28,31)(21,36,25,38,37,45,40,41)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3516);