C₅×#12389 #11935109

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

Aliases

direct product

C₅×#12389

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂×C₅×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#446 ◁ G

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

normal subgroups only

This group has 284 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)(22,23,27,31)(24,32,34,50)(25,36,38,52)(26,41,39,30)(28,43,44,33)(29,46,48,40)(35,51,42,47)(37,49,53,45);(9,18)(12,17)(14,21)(16,19)(22,24)(23,28)(25,37)(26,42)(27,34)(29,47)(30,49)(31,44)(32,36)(33,40)(35,39)(38,53)(41,45)(43,46)(48,51)(50,52);(6,8,10,15)(7,11,13,20)(9,19,18,16)(12,21,17,14)(25,38)(29,48)(32,50)(33,43)(36,52)(37,53)(40,46)(47,51)

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, 5377);