C₅×#8464 #11931184

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8453
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#8464

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#379
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
prime radical	C₅×#3487	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ G

1002 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1002 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)(8,12,13,10)(9,11)(14,15,19,24,20,25,18,23)(16,26,30,21,29,36,28,35)(17,22,31,37,32,38,33,39)(27,41,42,45,43,40,44,34);(6,8)(7,10)(9,13)(11,12)(14,16)(15,21)(17,27)(18,30)(19,28)(20,29)(22,34)(23,36)(24,26)(25,35)(31,44)(32,43)(33,42)(37,40)(38,45)(39,41);(14,17,20,32)(15,22,25,38)(16,27,29,43)(18,33,19,31)(21,34,35,45)(23,39,24,37)(26,40,36,41)(28,44,30,42)

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1452);