C₅×#10638 #11933358

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#10638

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

330 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 29
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,3,4,5)(6,7,9,12,8,10,13,11)(14,15,17,21,18,22,25,19)(16,23,27,29,24,28,26,20);(7,10)(11,12)(15,19)(16,24)(17,25)(20,23)(21,22)(28,29);(7,11)(9,13)(10,12)(14,16,25,27,18,24,17,26)(15,20,19,23,22,29,21,28)

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3626);