C₅×#10566 #11933286

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#10566

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#394
commutator, Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

490 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 114 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)(8,10)(9,11)(12,13)(14,15,19,23)(16,21,25,29)(17,26,24,20)(18,28,27,22);(6,8)(7,10)(9,12)(11,13)(14,16,19,25)(15,20,23,26)(17,27,24,18)(21,22,29,28);(10,13)(14,17,18,16,19,24,27,25)(15,21,22,26,23,29,28,20)

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, 3554);