C₅×#12889 #11935609

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#12889

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 49
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 49 points:

(1,2,3,4,5)(6,7,8,9)(10,11)(12,15)(13,14)(16,17)(18,19)(20,32)(21,26)(22,40)(23,43)(24,44)(25,31)(27,36)(28,33)(29,42)(30,41)(34,46)(35,45)(37,49)(38,48)(39,47);(11,14)(15,17)(18,20,33,38,24,35,42,39,25,36,23,34,41,37,22,21)(19,26,40,49,30,46,43,27,31,47,29,45,44,48,28,32);(10,12,13,16)(11,15,14,17)(18,21,22,37,41,34,23,36,25,39,42,35,24,38,33,20)(19,27,28,46,44,49,29,26,31,32,43,48,30,45,40,47)

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, 5877);