C₅×#8792 #11931512

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₅×#8792

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 53
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 53 points:

(1,2,3,4,5)(6,7)(8,13)(9,18)(10,20)(11,15)(12,16)(14,19)(17,21)(22,23)(24,32,35,28)(25,36,38,51)(26,31)(27,30)(29,37,42,52)(33,46)(34,45,49,41)(39,43)(40,47)(44,53)(48,50);(6,8,10,16,11,17,19,9)(7,12,14,13,15,18,20,21)(22,24,26,34,27,35,39,49)(23,28,31,41,30,32,43,45)(25,33,37,47,38,48,52,53)(29,40,36,50,42,44,51,46);(6,9,19,17,11,16,10,8)(7,13,20,12,15,21,14,18)(22,25,27,38)(23,29)(24,33,35,48)(26,37,39,52)(28,40)(30,42)(31,36)(32,44)(34,47,49,53)(41,50)(43,51)(45,46)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1780);