#72319087

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72285335
commutator	C₂×C₁₆	C₂²×C₄²
Frattini, cosocle	C₄×M₃₂	C₂⁴
prime radical	#7623252	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ #479908 ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1700 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=eh,c^2=fg,d^2=ik,e^2=h,f^2=i,i^2=j,j^2=k,c^a=cf,c^b=cik,d^a=dhi,d^b=dj,d^c=dhk,e^c=ejk,e^d=ek,f^a=fi,f^b=fj,f^c=fh,f^d=fk,f^e=fk,g^a=gh,g^c=gh,g^f=gk,h^c=hk,i^a=ij,i^b=ik,j^a=jk

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319398, #72343292, #72343317, …
maximal normal subgroup of	#72319398, #72343292, #72343317, …

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