C₅×#11676 #11934396

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#11676

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×#102
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

316 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 122 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)(6,7,9,12,10,8,13,11)(14,15)(16,25)(17,22)(18,31)(19,35)(20,24)(21,27)(23,33)(26,39)(28,45)(29,42)(30,40)(32,38)(34,41)(36,43)(37,44);(6,8)(7,10)(9,11)(12,13)(14,16,20,29)(15,21,24,37)(17,28,19,26)(18,32,34,43)(22,38,35,36)(23,39,40,45)(25,41,42,31)(27,30,44,33);(7,8)(11,12)(14,17,20,19)(15,22,24,35)(16,26,29,28)(18,30,34,33)(21,36,37,38)(23,31,40,41)(25,39,42,45)(27,43,44,32)

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 4664);