C₅×#12458 #11935178

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12430
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12458

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈×Q₈ ◁ G

192 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 53
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(22,23,27,31)(24,32,36,48)(25,37,39,52)(26,40,38,30)(28,41,43,34)(29,44,45,33)(35,51,50,47)(42,49,46,53);(22,24,33,49,26,35,39,41,27,36,44,53,38,50,25,34)(23,28,37,47,30,42,45,48,31,43,52,51,40,46,29,32);(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)

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, 5446);