C₅×#7502 #11930222

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7502

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#354
commutator	C₄²	C₄²×C₅
Frattini, cosocle	C₂×#124	C₂²×C₅
prime radical	C₂²×C₅×D₄	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#124 ◁ G

286 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 41 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 490);