C₅×#12932 #11935652

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×#12932

Subgroups

	subgroup	G/H
center	C₄×C₅	#481
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂×C₅	#1853
prime radical	C₂⁴×C₅	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

402 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 146 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)(6,7,10,14,11,16,26,15)(8,17,20,35,19,34,25,31)(9,21,24,13,23,33,36,32)(12,27,29,18,28,22,30,37);(6,8,11,19)(7,12)(9,22,23,27)(10,25,26,20)(13,31)(14,30)(15,29)(16,28)(17,21)(18,36,37,24)(32,35)(33,34);(9,23)(13,32)(17,34)(18,37)(21,33)(22,27)(24,36)(31,35)

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, 5920);