C₅×#7061 #11929781

order	1280
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#7061

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂²×C₄	C₂×C₅×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₅
socle	C₂²×C₅	#323
prime radical	C₂³×C₅	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

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

normal subgroups only

This group has 188 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) = 640

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

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

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 49);