C₅×#7526 #11930246

order	1280
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7526

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1404
commutator	#125	C₂×C₄×C₅
Frattini, cosocle	#491	C₂²×C₅
prime radical	C₂²×C₅	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#491 ◁ G

126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 514);