C₅×#12508 #11935228

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₅×#12508

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₅×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×C₈² ◁ G

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

normal subgroups only

This group has 280 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)(7,10)(8,12)(11,13)(15,19)(16,23)(17,24)(20,25)(21,26)(22,29)(27,28);(14,15,20,29,17,22,16,19,18,21,23,28,24,27,25,26);(6,7,8,10,9,11,12,13)(15,21)(19,26)(22,27)(28,29)

Conjugacy classes & character table

230 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5496);