C₅×#7524 #11930244

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×#7524

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1404
commutator	#125	C₂×C₄×C₅
Frattini, cosocle	#490	C₂²×C₅
prime radical	C₅×#127	D₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#490 ◁ G

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

normal subgroups only

This group has 134 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)(8,16)(9,20,22,13)(10,15)(11,14,24,32)(12,17,27,35)(18,34)(19,33,25,37)(21,28,36,29)(23,30,26,31);(6,8,10,18)(7,12,15,29)(9,21,23,17)(11,25,24,19)(13,16,31,34)(14,28,32,27)(20,37,30,33)(22,35,26,36)

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, 512);