C₅×#7525 #11930245

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×#7525

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1404
commutator	#125	C₂×C₄×C₅
Frattini, cosocle	#491	C₂²×C₅
prime radical	C₅×#562	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#491 ◁ G

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

normal subgroups only

This group has 138 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)(6,7,9,13,10,15,21,31)(8,16,18,29,17,27,34,36)(11,14,25,32,23,33,22,30)(12,20,26,24,28,35,37,19);(7,12)(8,17)(11,23)(13,29)(14,27)(15,28)(16,33)(20,35)(22,25)(26,32)(30,37)(31,36)

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, 513);