C₅×#12034 #11934754

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12034

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

204 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 118 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)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,33)(25,30)(26,40)(27,42)(28,32)(29,36)(31,34)(35,46)(37,52)(38,48)(39,45)(41,53)(43,50)(44,51)(47,49);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,34,49,27,37,41,51,28,38,39,50,25,35,26,36)(23,29,40,46,30,43,45,48,32,44,53,52,42,47,31,33);(8,14)(11,17)(15,20)(18,21)(22,25,28,27)(23,30,32,42)(24,35,38,37)(26,39,41,34)(29,43,44,47)(31,40,45,53)(33,46,48,52)(36,50,51,49)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5022);