Finite Group Databasebeta

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#10624710

order	729
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃²×C₂₇ · Schur multiplier (H₂) = C₃×C₉

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₉²	C₃²
commutator	C₃	C₃²×C₂₇
Frattini, cosocle	C₃×C₉	C₃³
socle	C₃²	C₃²×C₉
prime radical	C₃³	C₃×C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉² ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 243

Polycyclic presentation:

3^6|a^3=e,c^3=d,e^3=f,b^a=bd

Conjugacy classes & character table

297 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#13090967, #13090968, #13090969
maximal subgroup of	C₂×#10624710, #13090967, #13090968, …
maximal normal subgroup of	C₂×#10624710, #13090967, #13090968, …
maximal quotient of	C₂×#10624710, #14550234, C₉×C₃²⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(729, 270);