C₅×#8795 #11931515

order	1280
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅ · Schur multiplier (H₂) = C₈

Aliases

direct product

C₅×#8795

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#382
commutator	C₄²	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
prime radical	C₂³×C₅	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 320

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

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

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, 1783);