C₅×#19074 #11941794

order	1280
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₅×#19074

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3549	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

1072 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 1072 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 29 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 12062);