C₅×#7409 #11930129

order	1280
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7409

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₄×C₅
Frattini, cosocle	#391	C₂²×C₅
prime radical	C₅×#537	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#391 ◁ G

268 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

155 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 397);