C₅×#7420 #11930140

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7420

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#357
commutator	C₂×C₁₆	C₂×C₄×C₅
Frattini, cosocle	C₂²×C₁₆	C₂²×C₅
prime radical	C₅×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈ ◁ C₂²×C₅×C₁₆ ◁ G

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

normal subgroups only

This group has 150 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) = 640

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

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

Conjugacy classes & character table

245 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 408);