#72343163

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#392828
commutator	C₂×#131	C₂³×C₄
Frattini, cosocle	C₄×#131	C₂⁴
socle	C₂	#72343141
prime radical	#6469078	C₂²
LCS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

2204 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:

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

Polycyclic presentation:

2^11|a^2=e,c^2=h,d^2=g,e^2=i,h^2=jk,i^2=k,j^2=k,c^a=cik,d^a=df,d^b=dg,d^c=dfhij,e^c=ek,f^b=fijk,f^c=fk,f^d=fij,g^a=gij,g^c=gij,g^f=gk,h^a=hk,h^d=hjk,j^d=jk

Conjugacy classes & character table

122 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, 3, F.8);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.10*F.11);;
gap> SetPower(coll, 9, F.11);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 3, 1, F.3*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.11);;
gap> SetConjugate(coll, 6, 2, F.6*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9*F.10);;
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);