C₅×#7220 #11929940

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7217
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7220

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#340
commutator	C₂×C₈	C₄²×C₅
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
prime radical	C₂³×C₅	#103
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

188 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 90 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)(8,14,16,20)(9,12)(10,13)(11,15,18,21)(17,19)(22,23,26,32,29,34,28,44)(24,35,37,50,39,51,42,46)(25,33,27,43,41,49,40,31)(30,38,45,36,48,52,47,53);(6,8)(7,11,13,18)(9,15)(10,16)(12,14,19,20)(17,21)(22,24,27,38,26,42,41,53,29,39,40,52,28,37,25,36)(23,30,33,46,44,45,31,35,34,48,49,50,32,47,43,51)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 208);