#72227629

order	5120
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	H(9,𝔽₂) ⋊ D₅
derived	#72210028 ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂	#72209701
commutator, cosocle, hypocenter	#72210028	C₂
Fitting	H(9,𝔽₂)	D₅
LCS	G ▷ #72210028
UCS	C₁ ◁ C₂

At least 6659 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has at least 6659 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,9,17,24,6,2,14,7)(3,8,4,10,13,16,15,25)(5,21,31,22,20,12,32,18)(11,23,26,19,27,30,28,29);(2,8,24,30,31)(3,12,25,26,4)(5,11,22,29,14)(7,23,32,9,16)(10,28,15,13,21)(17,20,27,18,19)

Polycyclic presentation:

2,5,2^9|g^2=k,h^2=k,i^2=k,j^2=k,b^a=b^4,c^a=cf,c^b=d,d^a=cef,d^b=cde,e^a=cd,e^b=f,f^b=cd,g^a=cdik,g^b=h,g^c=gk,g^f=gk,h^a=fhk,h^b=i,h^d=hk,h^e=hk,h^g=hk,i^a=egk,i^b=j,i^e=ik,i^f=ik,i^h=ik,j^a=dfghij,j^b=ghijk,j^c=jk,j^e=jk,j^f=jk,j^i=jk

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227651, #72265383, #72272661
maximal normal subgroup of	#72265383, #72272661
maximal quotient of	#72268128, #72268211, #72268536, …

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", "k");;
gap> coll := SingleCollector(F, [2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2^4);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.4);;
gap> SetConjugate(coll, 4, 1, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.3*F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 6, 2, F.3*F.4);;
gap> SetConjugate(coll, 7, 1, F.3*F.4*F.9*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.6*F.8*F.11);;
gap> SetConjugate(coll, 8, 2, F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.5*F.7*F.11);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 5, F.9*F.11);;
gap> SetConjugate(coll, 9, 6, F.9*F.11);;
gap> SetConjugate(coll, 9, 8, F.9*F.11);;
gap> SetConjugate(coll, 10, 1, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 10, 2, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 3, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.10*F.11);;
gap> SetConjugate(coll, 10, 6, F.10*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);