#72210028

order	2560
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(9,𝔽₂) ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	C₂⁴⋊(C₂⁴⋊C₅)
commutator, Fitting, cosocle, hypocenter	H(9,𝔽₂)	C₅
LCS	G ▷ H(9,𝔽₂)
UCS	C₁ ◁ C₂

84286 conjugacy classes of subgroups; only the 21 normal subgroups are shown.

normal subgroups only

This group has 84286 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

5,2^9|d^2=j,e^2=j,f^2=j,g^2=j,b^a=c,c^a=bcd,c^b=cj,d^a=e,e^a=bc,e^b=ej,e^d=ej,f^a=g,f^b=fj,f^d=fj,g^a=fgh,g^b=gj,g^c=gj,g^e=gj,h^a=i,h^d=hj,h^e=hj,h^f=hj,i^a=fg,i^b=ij,i^d=ij,i^e=ij,i^g=ij,i^h=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227629, #72227762, C₂×#72210032, …
maximal subgroup of	#72210032, #72210041, #72227629, …
maximal normal subgroup of	#72210032, #72210041, #72227629, …
maximal quotient of	#72241456, C₂×#72210028, #72268127, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [5, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 5, 1, F.2*F.3);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.6*F.7);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);