#72311271

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,11)(5,8)(7,12)(9,13)(10,14)(15,16)(17,18)(19,24)(20,25)(21,33)(22,36)(23,28)(26,35)(27,38)(29,39)(30,40)(31,41)(32,42)(34,44)(37,43)(45,48)(46,47);(1,2)(3,6)(4,7,12,11)(5,8)(9,15,16,13)(10,14)(17,19,23,20)(18,24,28,25)(21,31,35,29)(22,32)(26,39,33,41)(27,42)(30,38)(34,46)(36,40)(37,45)(43,47)(44,48);(1,3,5,10)(2,6,8,14)(4,9)(7,15)(11,13)(12,16)(17,20,23,19)(18,25,28,24)(21,31,35,29)(22,32)(26,39,33,41)(27,40)(30,38)(34,46)(36,42)(37,45)(43,48)(44,47);(1,4)(2,7,8,11)(3,9)(5,12)(6,13,14,15)(10,16)(17,21,23,35)(18,26)(19,29)(20,31)(22,37,38,34)(24,39,25,41)(27,44)(28,33)(30,46)(32,45)(36,43)(40,48,42,47);(3,10)(4,12)(7,11)(13,15)(17,22)(18,27)(19,30)(20,32)(21,34)(23,38)(24,40)(25,42)(26,43)(28,36)(29,45)(31,46)(33,44)(35,37)(39,47)(41,48)

Polycyclic presentation:

2^10|b^2=fj,c^2=fij,d^2=ghj,b^a=bf,d^a=dfh,d^b=dfg,d^c=dgij,e^a=ehj,e^b=eh,e^c=eghij,e^d=eij,f^d=fi,f^e=fj,g^a=gij,g^b=gi,g^c=gi,g^d=gj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311321, #72340457, #72340480
maximal normal subgroup of	#72311321, #72340457, #72340480

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