C₅×#7082 #11929802

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7082

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#327
commutator	C₂²×C₈	C₂×C₄×C₅
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

204 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 320

A minimal faithful permutation representation — 2 generators on 45 points:

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

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, 70);