#72285373

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

2793 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)(3,10,13,7)(4,14)(5,8)(6,9)(11,16,15,12)(17,18)(19,29,33,24)(20,25,36,43)(21,26)(22,27,35,45)(23,28)(30,41)(31,39,48,44)(32,46)(34,38)(37,42)(40,47);(1,3,6,13)(2,7,9,10)(4,15,8,11)(5,12,14,16)(17,19)(18,24)(20,34)(21,31)(22,32)(23,33)(25,41)(26,44)(27,40)(28,29)(30,36)(35,47)(37,48)(38,43)(39,42)(45,46);(1,4,6,8)(2,5)(3,11,13,15)(7,16)(9,14)(10,12)(17,20)(18,25,28,43)(19,30)(21,22)(23,36)(24,38,29,41)(26,45,42,27)(31,47)(32,48)(33,34)(35,37)(39,46,44,40);(1,5,6,14)(2,8,9,4)(3,12)(7,11)(10,15)(13,16)(17,21)(18,26)(19,31,33,48)(20,35,36,22)(23,37)(24,39,29,44)(25,27,43,45)(28,42)(30,47)(32,34)(38,46)(40,41);(1,5)(2,8,9,4)(3,12)(6,14)(7,15,10,11)(13,16)(17,22,23,35)(18,27,28,45)(19,32,33,47)(20,21)(24,40,29,46)(25,42)(26,43)(30,31)(34,48)(36,37)(38,39)(41,44)

Polycyclic presentation:

2^10|a^2=g,b^2=ij,c^2=fij,d^2=gij,e^2=ghj,b^a=bf,c^a=cfgh,c^b=ci,d^a=dg,d^b=dfghi,d^c=dhij,e^a=efghi,e^b=egh,e^c=efhj,e^d=ehij,f^a=fi,f^d=fij,f^e=fij,g^b=gi,g^c=gj,g^e=gj,h^a=hij,h^b=hi,h^c=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526
maximal normal subgroup of	#72285526

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