#72286347

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235715
commutator, Frattini, cosocle	#2309	C₂⁴
prime radical	#72285797	C₂
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35272 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^11|a^2=i,b^2=gk,c^2=hik,d^2=ik,e^2=k,g^2=ik,b^a=bg,c^a=cjk,c^b=cf,d^a=dik,d^c=de,e^a=ek,e^b=ei,e^c=ek,f^a=fh,f^b=fhj,f^c=fj,f^d=fik,f^e=fk,g^a=gik,g^c=ghj,g^f=gjk,h^b=hj,h^c=hk,h^g=hk,i^c=ik,j^a=jk,j^b=jk

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286380
maximal normal subgroup of	#72286380

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