C₅×#12388 #11935108

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12388

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#3403	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#446 ◁ G

296 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 78 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)(6,7,10,13)(8,14,15,21)(9,17,18,12)(11,19,20,16)(22,23,28,32)(24,33,36,51)(25,30,27,42)(26,38,41,49)(29,44,46,35)(31,37,50,40)(34,52,43,48)(39,45,53,47);(9,18)(12,17)(14,21)(16,19)(22,24)(23,29)(25,34)(26,39)(27,43)(28,36)(30,45)(31,48)(32,46)(33,38)(35,40)(37,44)(41,53)(42,47)(49,51)(50,52);(6,8,10,15)(7,11,13,20)(9,19,18,16)(12,21,17,14)(22,25,28,27)(23,30,32,42)(24,34,36,43)(26,40,41,37)(29,45,46,47)(31,49,50,38)(33,48,51,52)(35,53,44,39)

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5376);