#72285893

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2845 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)(4,7)(5,12)(8,15)(9,16)(10,11)(13,14)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,39)(29,38)(31,36)(32,40)(33,42)(34,41)(35,43)(37,46)(44,48)(45,47);(1,3)(2,6)(4,11)(5,13,15,9)(7,10)(8,16,12,14)(17,19)(18,23)(20,31)(21,29,35,46)(22,30)(24,39)(25,37,43,38)(26,27)(28,33)(32,44,34,45)(36,42)(40,47,41,48);(1,4,7,2)(3,6,10,11)(5,12)(8,15)(9,16)(13,14)(17,20)(18,24)(19,28)(21,34,35,32)(22,33)(23,36)(25,40,43,41)(26,42)(27,39)(29,45,46,44)(30,31)(37,47,38,48);(1,5)(2,8,4,12)(3,9)(6,14,11,16)(7,15)(10,13)(17,21)(18,25,26,43)(19,29,30,46)(20,32)(22,35)(23,37)(24,40,42,41)(27,38)(28,44,31,45)(33,34)(36,47)(39,48);(1,6)(2,3,4,10)(5,14,15,16)(7,11)(8,9)(12,13)(17,21)(18,25,26,43)(19,29)(20,32,33,34)(22,35)(23,38,27,37)(24,41)(28,45,31,44)(30,46)(36,48)(39,47)(40,42)

Polycyclic presentation:

2^10|b^2=f,c^2=fj,d^2=gh,e^2=g,b^a=bij,c^a=cfj,c^b=cfi,d^a=dfij,d^b=dfh,d^c=dg,e^a=efhi,e^b=eghij,e^c=efhi,e^d=eh,f^d=fi,f^e=fij,g^a=gi,g^b=gj,g^c=gi,g^d=gj,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286044, #72340228, #72340293, …
maximal normal subgroup of	#72286044, #72340228, #72340293, …

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