C₅×#12076 #11934796

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅×#12076

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

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

normal subgroups only

This group has 210 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)(6,7,9,8,10,11,13,12)(14,15,18,22)(16,24,25,43)(17,28,31,42)(19,33,30,23)(20,35,36,26)(21,38,40,32)(27,44,34,39)(29,41,45,37);(14,16,18,25)(15,20,22,36)(17,29,31,45)(19,34,30,27)(21,39,40,44)(23,41,33,37)(24,42,43,28)(26,38,35,32);(6,8,9,11,10,12,13,7)(16,25)(20,36)(24,43)(26,35)(27,34)(29,45)(37,41)(39,44)

Conjugacy classes & character table

290 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5064);