C₅×#12771 #11935491

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₅×#12771

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1790
commutator	C₂×D₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×#41	C₂³×C₅
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ G

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

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5759);