C₅×#7418 #11930138

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7418

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

162 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 54 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)(8,12,13,10)(9,11)(14,15)(16,20)(17,29)(18,32)(19,24)(21,25)(22,26)(23,30)(27,39)(28,35)(31,45)(33,41)(34,40)(36,44)(37,42)(38,43);(6,8)(7,10)(9,13)(11,12)(15,20)(16,25)(17,26)(18,30)(21,22)(23,39)(24,35)(27,44)(28,42)(29,33)(31,43)(32,34)(36,45)(37,38)(40,41)

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, 406);