C₅×#7405 #11930125

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7405

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#353
commutator	C₂×C₄²	C₂×C₄×C₅
Frattini, cosocle	#391	C₂²×C₅
prime radical	C₂³×C₅	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#391 ◁ G

252 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

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

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, 393);