C₅×#7452 #11930172

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#7452

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#357
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₄×C₁₆	C₂²×C₅
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

102 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 2 generators on 45 points:

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 440);