C₅×#12297 #11935017

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12297

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#424 ◁ G

290 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 53 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5285);