C₅×#12876 #11935596

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12876

Subgroups

	subgroup	G/H
center	C₄×C₅	#481
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1824
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

370 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

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

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, 5864);