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#13015355

order	1280
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

C₅ ⋊ #7867

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄²×C₅	C₄
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²⋊C₄
cosocle	C₂×C₄×Dic₅	C₂³
hypercenter	C₂²×C₄²	AGL(1,𝔽₅)
hypocenter	C₅	#7867
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄²

922 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 160

Polycyclic presentation:

2^8,5|a^2=e,b^2=f,d^2=g,e^2=h,b^a=bd,c^a=ch,d^a=dg,f^a=fg,i^a=i^2,i^e=i^4

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 1085623);