#72231575

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#6453969
commutator, Frattini, cosocle	C₂³×C₈	C₂⁵
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#60207 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

At least 8302 conjugacy classes of subgroups; only the 1184 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(7,15)(16,31)(17,32)(19,33);(5,6)(7,11,15,26)(8,13)(9,12)(10,14)(16,23,31,36)(17,34,32,24)(18,28)(19,27,33,35)(20,30)(21,29)(22,25);(7,16)(9,22)(10,21)(11,23)(12,25)(14,29)(15,31)(17,19)(18,20)(24,35)(26,36)(27,34)(28,30)(32,33);(2,4)(7,17,15,32)(9,18)(10,21)(11,24,26,34)(12,28)(14,29)(16,19,31,33)(20,22)(23,35,36,27)(25,30);(1,2,3,4)(5,7,18,17,10,19,9,16,8,15,22,32,21,33,20,31)(6,11,25,24,14,27,30,36,13,26,28,34,29,35,12,23)

Polycyclic presentation:

2^11|b^2=i,d^2=i,e^2=fgj,g^2=jk,j^2=k,b^a=bi,d^b=di,e^a=ef,e^b=eh,e^c=eg,e^d=efgj,f^b=fk,g^c=gj,g^d=gjk,h^b=hk,h^e=hk,i^e=ik,j^c=jk,j^d=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256975
maximal normal subgroup of	#72256975
automorphism group of	#1381

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