C₅×#12845 #11935565

order	1280
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12845

Subgroups

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂³	C₂²×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

454 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 49
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5)(6,7)(8,9)(10,11)(12,15)(13,14)(16,17)(18,19,21,25,22,26,31,37,23,27,32,38,33,39,43,47)(20,24,29,35,30,36,42,46,28,34,40,44,41,45,48,49);(11,14)(15,17)(18,20)(19,24)(21,29)(22,30)(23,28)(25,35)(26,36)(27,34)(31,42)(32,40)(33,41)(37,46)(38,44)(39,45)(43,48)(47,49);(7,9)(10,12,13,16)(11,15,14,17)(20,28)(24,34)(29,40)(30,41)(35,44)(36,45)(42,48)(46,49)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5833);