C₅×#7445 #11930165

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

Aliases

direct product

C₅×#7445

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

132 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 58 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)(6,7,8,11,9,10,12,13)(15,20)(16,24)(17,26)(18,30)(21,36)(22,39)(23,35)(25,29)(27,43)(28,33)(31,37)(32,34)(38,44)(40,41)(42,45);(7,10)(8,12)(14,15,19,23)(16,25,28,42)(17,29,31,45)(18,32,30,22)(20,33,35,24)(21,37,38,26)(27,44,40,36)(34,41,39,43)

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, 433);