C₅×#11752 #11934472

order	1280
nilpotency class	4
isoclinism	isoclinic class of #11704
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#11752

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₂³×C₅	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

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

normal subgroups only

This group has 320 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) = 45
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4740);