C₅×#10028 #11932748

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10026
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10028

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#3245	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

512 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 37
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,3,4,5)(6,7)(8,15)(9,20)(10,13)(11,14)(12,17)(16,19)(18,21)(22,23)(24,30)(25,33)(26,29)(27,32)(28,35)(31,37)(34,36);(6,8,11,19)(7,12,14,21)(9,18,10,17)(13,16,20,15)(27,36)(30,37);(6,7)(8,16)(9,13)(10,20)(11,14)(12,17)(15,19)(18,21)(22,24)(23,27)(25,34)(26,31)(28,37)(29,36)(30,33)(32,35)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3016);