#72310896

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

2799 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=j,e^2=ij,b^a=bf,d^a=dh,d^c=dgj,e^a=eh,e^b=eh,e^c=efg,f^b=fi,f^d=fj,f^e=fj,g^a=gi,g^e=gj,h^b=hj,h^c=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72340329, #72340400, …
maximal normal subgroup of	#72310969, #72340329, #72340400, …

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