C₅×#7194 #11929914

order	1280
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅×#7194

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#99
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂²×M₁₆	C₂²×C₅
socle	C₂²×C₅	#323
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×M₁₆ ◁ G

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 41
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 41 points:

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

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 182);