C₅×#7505 #11930225

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7505

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#354
commutator	C₄²	C₄²×C₅
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₅
prime radical	C₂³×C₅	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊Q₈ ◁ G

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

normal subgroups only

This group has 260 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)(10,11,13,18,15,20,27,37)(12,21,23,38,22,33,39,35)(14,19,28,36,29,17,26,34)(16,30,31,24,32,25,41,40);(6,7,8,9)(10,12,14,24,15,22,29,40)(11,16,19,21,20,32,17,33)(13,25,26,23,27,30,28,39)(18,35,34,41,37,38,36,31)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 493);