C₅×#8477 #11931197

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×#8477

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₅×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

682 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 142 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)(6,7,9,13,10,12,18,20)(8,14,15,21,16,11,17,19)(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);(6,8,10,16)(7,11,12,14)(9,17,18,15)(13,21,20,19)(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);(7,12)(9,18)(11,14)(15,17)(22,25)(23,29)(24,33)(26,41)(27,38)(28,43)(30,47)(31,36)(32,48)(34,51)(35,50)(37,42)(39,45)(40,49)(44,52)(46,53)

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, 1465);