C₅×#12591 #11935311

order	1280
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅×#12591

Subgroups

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

390 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 33
transitive degree	μ_t(G) = 1280

A minimal faithful permutation representation — 3 generators on 33 points:

(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,20,24,21,25,26,31,22,23,27,29,28,30,32,33);(6,7,8,9)(11,14)(15,17);(10,12,13,16)(11,15,14,17)(19,23)(24,29)(25,30)(31,33)

Conjugacy classes & character table

560 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5579);