C₅×#11862 #11934582

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11862

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3169	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

348 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 118 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)(8,14,16,11)(9,12)(10,13)(15,20,17,19)(18,21)(22,23,26,31,28,33,27,32)(24,34,25,37,36,30,35,29);(6,8)(7,11)(9,17)(10,16)(12,20)(13,14)(15,18)(19,21)(23,29)(24,25)(26,27)(30,31)(32,34)(33,37)(35,36);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,24,27,35,28,36,26,25)(23,29,31,34,33,37,32,30)

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, 4850);