C₅×#9007 #11931727

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#9007

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#120
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂²×C₅	#386
prime radical	C₅×#3150	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

502 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 640

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 1995);