C₅×#12286 #11935006

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#12286

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂×C₅×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#408 ◁ G

306 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 74 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)(8,11,15,18)(9,12,16,19)(14,20,21,17)(23,29)(24,33)(25,36)(26,27)(30,44)(31,48)(32,43)(34,50)(35,40)(37,41)(38,39)(42,45)(46,53)(47,51);(6,8)(7,11)(9,14)(10,15)(12,17)(13,18)(16,21)(19,20)(22,23,28,32)(24,30,35,47)(25,37,39,48)(26,31,27,41)(29,38,43,36)(33,45,40,46)(34,51,50,44)(42,52,53,49);(6,9)(7,12)(8,14)(10,16)(11,17)(13,19)(15,21)(18,20)(22,24)(23,30)(25,34)(26,40)(27,33)(28,35)(29,42)(31,46)(32,47)(36,49)(37,51)(38,52)(39,50)(41,45)(43,53)(44,48)

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, 5274);