C₅×#7072 #11929792

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7072

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#102
commutator	C₂×C₈	C₂×C₅×C₈
Frattini, cosocle	C₈²	C₂²×C₅
socle	C₂²×C₅	D₄⋊C₈
prime radical	C₂×C₅×D₈	C₈
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₈² ◁ G

146 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 37 points:

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

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, 60);