#72343205

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₄×#1910
commutator	C₂×M₁₆	C₂²×C₄²
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72343130
prime radical	#6469078	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₁₆ ◁ #33326 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2755 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:

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

Polycyclic presentation:

2^11|a^2=e,b^2=h,d^2=ghjk,e^2=i,g^2=j,h^2=j,i^2=k,j^2=k,b^a=bjk,c^a=ck,c^b=cij,d^b=dg,d^c=df,e^b=ek,f^b=fijk,f^d=fijk,g^a=gk,g^c=gijk,g^d=gjk,g^f=gk,h^a=hk,h^d=hjk,j^d=jk

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343240
maximal normal subgroup of	#72343240

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