C₅×#12593 #11935313

order	1280
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅×C₈ · Schur multiplier (H₂) = C₄²

Aliases

direct product

C₅×#12593

Subgroups

	subgroup	G/H
center	C₂×C₅×C₈	C₄²
commutator	C₄	C₂×C₄×C₅×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₈ ◁ G

338 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 222 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)(6,7)(8,11)(9,10)(12,13)(14,15,19,25,20,27,35,42,21,26,34,41,18,24,17,23)(16,28,31,44,32,22,29,38,33,40,36,37,30,39,45,43);(7,10)(11,13)(14,16)(15,22,26,39)(17,29)(18,30)(19,36)(20,32)(21,33)(23,37,42,44)(24,28,27,40)(25,43,41,38)(31,34)(35,45);(6,8,9,12)(7,11,10,13)(14,17,18,34,21,35,20,19)(15,23,24,41,26,42,27,25)(16,29,30,31,33,45,32,36)(22,37,28,38,39,44,40,43)

Conjugacy classes & character table

440 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5581);