C₅×#7202 #11929922

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×#7202

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₄×C₅×C₈
Frattini, cosocle	C₂³×C₈	C₂²×C₅
socle	C₂²×C₅	#336
prime radical	C₂⁴×C₅	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂³×C₅×C₈ ◁ G

304 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 320

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

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

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, 190);