C₅×#13834 #11936554

order	1280
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄²×C₅ · Schur multiplier (H₂) = C₂×C₄³

Aliases

direct product

C₅×#13834

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅	C₂⁴
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂⁴	C₂⁴×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

1390 conjugacy classes of subgroups; only the 622 normal subgroups are shown.

normal subgroups only

This group has 1390 conjugacy classes of subgroups — too many to draw. Only its 622 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) = 1280

A minimal faithful permutation representation — 4 generators on 29 points:

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

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, 6822);