C₅×#9780 #11932500

order	1280
nilpotency class	3
isoclinism	isoclinic class of #9758
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₈

Aliases

direct product

C₅×#9780

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#390
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

558 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 320

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

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 2768);