#72319283

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#354936
commutator	C₂²×C₈	C₂²×C₄²
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72318993
prime radical	#3444	#41
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄×C₈ ◁ #479961 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2128 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=egj,d^2=hjk,e^2=ijk,g^2=jk,h^2=j,j^2=k,b^a=bh,c^a=cf,d^a=dhjk,d^b=djk,d^c=dik,e^b=ek,e^c=ei,e^d=ek,f^a=fi,f^d=fk,f^e=fk,g^a=gj,g^b=gk,g^c=gi,g^f=gk,h^a=hj,h^b=hk,i^a=ik,j^a=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319398, #72343240, #72343292
maximal normal subgroup of	#72319398, #72343240, #72343292

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