C₅×#7518 #11930238

order	1280
nilpotency class	6
isoclinism	isoclinic class of #7515
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7518

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1402
commutator	#127	C₂×C₄×C₅
Frattini, cosocle	#492	C₂²×C₅
prime radical	C₅×#127	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#492 ◁ G

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

normal subgroups only

This group has 148 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)(6,7,9,14,11,16,23,32)(8,17,18,30,20,27,34,37)(10,13,21,29,24,15,22,31)(12,25,26,35,28,33,36,19);(6,8,11,20)(7,12,16,28)(9,18,23,34)(10,19,24,35)(13,17,15,27)(14,30,32,37)(21,25,22,33)(26,29,36,31)

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 506);