#13014282

order	1280
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#367 ⋊ C₄
Sylow	C₅ ⋊ #7423

Subgroups

	subgroup	G/H
center	C₂	#10598611
commutator	C₂×C₅×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×AGL(1,𝔽₅)
Fitting	C₅×#367	C₄
socle	C₂×C₅	#1347
prime radical	C₂²×D₅	D₁₆
cosocle	C₂×#3930	C₂²
hypercenter	#367	AGL(1,𝔽₅)
hypocenter	C₅	#7423
LCS	G ▷ C₂×C₅×C₁₆ ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

140 conjugacy classes of subgroups; only the 34 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 34 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 320

Polycyclic presentation:

2^8,5|a^2=d,b^2=h,c^2=fgh,e^2=g,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fgh,g^a=gh,i^a=i^2,c^b=cf,d^b=defgh,e^b=egh,f^b=fgh,g^b=gh,i^b=i^4,d^c=dh,i^d=i^4

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1084550);