C₅×#8632 #11931352

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#8632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#380
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

388 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 130 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)(6,7,8,11,9,10,12,13)(14,15)(16,25)(17,30,32,37)(18,33)(19,23)(20,24)(21,27)(22,31,38,34)(26,35,42,45)(28,36)(29,40)(39,41,44,43);(14,16,19,28,20,29,18,27)(15,21,33,40,24,36,23,25)(17,26,34,43,32,42,31,41)(22,35,30,44,38,45,37,39);(7,10)(8,12)(14,17)(15,22)(16,26)(18,31)(19,34)(20,32)(21,35)(23,37)(24,38)(25,39)(27,41)(28,43)(29,42)(30,33)(36,45)(40,44)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1620);