C₅×#7483 #11930203

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7481
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7483

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#1390
commutator	C₂×Q₈	C₄²×C₅
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₅
prime radical	C₂³×C₅	#99
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊Q₈ ◁ G

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

normal subgroups only

This group has 262 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)(6,7,9,14,11,15,23,33)(8,17,19,12,21,27,36,28)(10,16,25,13,26,30,22,31)(18,34,20,32,37,29,24,35);(6,8)(7,12,15,28)(9,18)(10,24)(11,21)(13,17,31,27)(14,32,33,35)(16,34,30,29)(19,22)(20,26)(23,37)(25,36)

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, 471);