C₅×#7471 #11930191

order	1280
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7471

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	D₁₆
commutator	C₁₆	C₄²×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 190 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 — 2 generators on 45 points:

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

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 459);