C₅×#12039 #11934759

order	1280
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12039

Subgroups

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

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

normal subgroups only

This group has 242 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) = 29
transitive degree	μ_t(G) = 640

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

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

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, 5027);