C₅×#7089 #11929809

order	1280
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7089

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#327
commutator	C₂²×C₈	C₂×C₄×C₅
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

180 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 320

A minimal faithful permutation representation — 2 generators on 37 points:

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

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, 77);