C₅×#8478 #11931198

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₅×#8478

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

656 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 640

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

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

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, 1466);