C₅×#12062 #11934782

order	1280
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12062

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂×D₄
commutator	C₄	C₂²×C₅×C₁₆
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₈×D₄
prime radical	C₂×C₅×D₈	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

252 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 29
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5050);