C₅×C₈×Q₃₂ #11934762

order	1280
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×C₈×Q₃₂

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	D₈
commutator	C₈	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂²×C₅	C₄×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₈ ◁ C₅×C₈² ◁ G

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

normal subgroups only

This group has 226 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) = 1280

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

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

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, 5030);