C₅×#19058 #11941778

order	1280
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#19058

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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)(6,7,10,13)(8,14,15,21)(9,17,18,12)(11,19,20,16)(22,23)(24,34)(25,39)(26,31)(27,32)(28,33)(29,37)(30,41)(35,43)(36,44)(38,47)(40,45)(42,46)(48,50)(49,52)(51,53);(9,18)(12,17)(14,21)(16,19)(22,24)(23,29)(25,35)(26,36)(27,37)(28,38)(30,43)(31,34)(32,47)(33,44)(39,52)(40,49)(41,50)(42,51)(45,53)(46,48);(6,8)(7,11)(9,19)(10,15)(12,21)(13,20)(14,17)(16,18)(22,25,28,42)(23,30,33,45)(24,35,38,51)(26,41,27,40)(29,43,44,53)(31,46,32,39)(34,48,47,52)(36,50,37,49);(22,26,28,27)(23,31,33,32)(24,36,38,37)(25,40,42,41)(29,34,44,47)(30,39,45,46)(35,49,51,50)(43,52,53,48)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 12046);