C₅×#32877 #11955597

order	1280
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#32877

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

594 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 37 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 25865);