C₅×#7393 #11930113

order	1280
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×#7393

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₅×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂²×C₅
socle, prime radical	C₂²×C₅	C₂²⋊C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₁₆ ◁ G

110 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 2 generators on 45 points:

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

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, 381);