#72310992

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
prime radical	#245252	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2789 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)(3,4)(5,13)(6,9)(7,11)(8,15)(10,12)(14,16)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,38)(28,39)(29,40)(31,44)(32,42)(35,46)(37,41)(43,48)(45,47);(1,3)(2,7)(4,9)(5,14)(6,11)(8,16)(10,15)(12,13)(17,19)(18,23)(20,31)(21,35)(22,29)(24,38)(25,39)(26,40)(27,33)(28,36)(30,44)(32,45)(34,46)(37,43)(41,48)(42,47);(1,4,6,7)(2,3,9,11)(5,12,15,16)(8,10,13,14)(21,36)(32,37)(39,46)(47,48);(1,5,6,15)(2,8)(3,10)(4,12,7,16)(9,13)(11,14)(17,20)(18,24,26,30)(19,27,29,31)(21,37,36,32)(22,33)(23,38)(25,42)(28,43)(34,41)(35,45)(39,47,46,48)(40,44);(3,11)(12,16)(17,21)(18,25)(19,28)(20,32)(22,36)(23,39)(24,41)(26,34)(27,43)(29,35)(30,42)(31,45)(33,37)(38,47)(40,46)(44,48)

Polycyclic presentation:

2^10|c^2=i,d^2=ghi,b^a=bi,c^a=cf,c^b=cf,d^a=dfh,d^b=dfg,e^a=ehj,e^b=efgij,e^c=efghj,e^d=eh,f^d=fi,f^e=fj,g^a=gij,g^c=gi,g^d=gj,g^e=gj,h^b=hij,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311029
maximal normal subgroup of	#72311029

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