C₅×#19616 #11942336

order	1280
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#19616

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
socle	C₂²×C₅	C₂×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄²×C₅ ◁ G

906 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 278 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 — 4 generators on 29 points:

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

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, 12604);