C₅×#7147 #11929867

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#7147

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#95
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂³×C₅	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

188 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 2 generators on 53 points:

(1,2,3,4,5)(6,7)(8,15,17,21)(9,13)(10,19)(11,14)(12,16,20,18)(22,23,25,29,26,31,36,43,27,30,37,42,38,44,49,52)(24,32,34,45,35,46,48,53,33,28,39,40,47,41,50,51);(6,8,10,18,11,17,9,16)(7,12,13,15,14,20,19,21)(22,24)(23,28,30,32)(25,34)(26,35)(27,33)(29,40,42,45)(31,41,44,46)(36,48)(37,39)(38,47)(43,51,52,53)(49,50)

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 135);