#10700330

order	768
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

	subgroup	G/H
center	C₂²	C₃⋊#401
commutator, prime radical, cosocle	C₂⁵×C₃	C₂³
Frattini	C₂⁵	C₂²×D₃
Fitting	C₃×#1304	C₂
socle	C₂²×C₃	#401
hypercenter	#1304	D₃
hypocenter	C₃	#10690
LCS	G ▷ C₂⁵×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1304

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

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

μ_t(G) = 96

Polycyclic presentation:

2^8,3|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,i^b=i^2,e^c=eg,f^c=fh,i^c=i^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 74711);