C₅×#18732 #11941452

order	1280
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#18732

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂³×C₅	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

808 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 53 points:

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

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, 11720);