C₅×#7289 #11930009

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7289

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#340
prime radical	C₂³×C₅	#106
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

184 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 640

A minimal faithful permutation representation — 2 generators on 53 points:

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

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 277);