#72310883

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

2963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=g,e^2=ij,c^a=cg,c^b=cf,d^b=dgj,e^a=ehi,e^b=ehj,e^c=ehj,e^d=efgj,f^a=fi,f^b=fj,f^c=fi,f^e=fj,g^b=gi,g^e=gj,h^c=hj,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310969, #72311043, #72340254, …
maximal normal subgroup of	#72310969, #72311043, #72340254, …

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