#72285276

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72235685
commutator, Frattini, cosocle	#2309	C₂⁴
LCS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×C₄ ◁ #35327 ◁ G
LECS	G ▷ #2309 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

4275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,4)(3,13,18,40)(5,10,26,19)(6,11,28,20)(7,12,30,21)(9,31,14,17)(15,38,22,51)(16,39,47,63)(23,27,35,24)(25,49,42,36)(29,37,57,50)(32,53,34,45)(33,58,41,48)(43,64,52,60)(44,62,46,61)(54,55,59,56);(1,3,7,17)(2,9,12,13)(4,14,21,40)(5,22,25,53,26,15,42,45)(6,16,29,48)(8,18,30,31)(10,34,36,51,19,32,49,38)(11,33,37,39)(20,41,50,63)(23,43,54,62,35,52,59,61)(24,44,55,60,27,46,56,64)(28,47,57,58);(1,4,8,2)(3,14,18,9)(5,23)(6,20,28,11)(7,21,30,12)(10,24)(13,17,40,31)(15,43)(16,41,47,33)(19,27)(22,52)(25,54)(26,35)(29,50,57,37)(32,44)(34,46)(36,55)(38,60)(39,48,63,58)(42,59)(45,62)(49,56)(51,64)(53,61);(1,5)(2,10)(3,15)(4,19)(6,27)(7,25)(8,26)(9,32)(11,23)(12,36)(13,38)(14,34)(16,46)(17,45)(18,22)(20,35)(21,49)(24,28)(29,56)(30,42)(31,53)(33,43)(37,54)(39,62)(40,51)(41,52)(44,47)(48,64)(50,59)(55,57)(58,60)(61,63)

Polycyclic presentation:

2^11|a^2=k,b^2=g,c^2=ik,e^2=k,g^2=i,b^a=bg,c^a=ci,c^b=ci,d^a=df,d^b=dfj,d^c=de,e^a=ei,e^b=ei,e^d=ek,f^b=fh,f^c=fi,f^e=fk,g^a=gi,g^d=ghk,g^f=gjk,h^a=hj,h^b=hjk,h^g=hk,i^d=ik,j^b=jk

Conjugacy classes & character table

62 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.11);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.9*F.11);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.10*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.9*F.11);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);