C₅×#12288 #11935008

order	1280
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₅×#12288

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅
prime radical	C₅×#3245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×C₄⋊C₈ ◁ G

354 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 37
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 5276);