C₅×#10218 #11932938

order	1280
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₅×#10218

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

650 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 29 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 3206);