C₅×#32876 #11955596

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

Aliases

direct product

C₅×#32876

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

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

normal subgroups only

This group has 584 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)(6,7,11,15)(8,16,18,12)(9,20,22,13)(10,24,26,14)(17,32,33,27)(19,28,25,31)(21,29,36,34)(23,30,37,35);(6,8,10,25,11,18,26,19)(7,12,14,31,15,16,24,28)(9,21,23,17,22,36,37,33)(13,29,30,27,20,34,35,32);(6,9,10,23,11,22,26,37)(7,13,14,30,15,20,24,35)(8,17,19,21,18,33,25,36)(12,27,28,29,16,32,31,34);(17,33)(21,36)(27,32)(29,34)

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, 25864);