C₅×#7186 #11929906

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7186

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂³×C₈	C₂²×C₅
socle	C₂³×C₅	C₂²⋊C₈
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

326 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 49
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 2 generators on 49 points:

(1,2,3,4,5)(10,11)(12,16,17,14)(13,15)(18,19,22,27,23,28,37,45,24,29,38,46,21,26,35,43)(20,30,32,47,31,41,48,44,33,42,36,49,34,25,39,40);(6,7,8,9)(10,12)(11,14)(13,17)(15,16)(18,20)(19,25)(21,34)(22,36)(23,31)(24,33)(26,42)(27,44)(28,30)(29,41)(32,38)(35,48)(37,39)(40,46)(43,47)(45,49)

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 174);