C₅×#7490 #11930210

order	1280
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅×#7490

Subgroups

	subgroup	G/H
center	C₄×C₅	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₅
Frattini, cosocle	#532	C₂²×C₅
socle	C₂×C₅	#1390
prime radical	C₅×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#532 ◁ G

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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)(6,7,9,14)(8,18,20,12)(10,15,25,32)(11,17,27,34)(13,28,16,26)(19,35,22,33)(21,36,24,29)(23,31,37,30);(6,8,10,21,11,23,28,22)(7,12,15,29,17,30,16,35)(9,24,25,37,27,19,26,20)(13,31,14,33,32,18,34,36)

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, 478);