C₅×#12504 #11935224

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12503
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12504

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

262 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 1280

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

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

Conjugacy classes & character table

350 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5492);