C₅×#8524 #11931244

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8524

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂²×C₅×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

910 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 150 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 — 3 generators on 45 points:

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

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, 1512);