C₅×#12085 #11934805

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12085

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂²×C₅	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₈² ◁ G

206 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 45
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 45 points:

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

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, 5073);