C₅×#7739 #11930459

order	1280
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#7739

Subgroups

	subgroup	G/H
center, socle	C₂³×C₅	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂×C₅×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ G

484 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 640

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 727);