#72285935

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

2788 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)(3,9)(4,12)(5,8)(6,11)(7,13)(10,14)(15,16)(17,18)(19,23)(20,24)(21,34)(22,27)(25,37)(26,33)(28,41)(29,39)(30,38)(31,40)(32,42)(35,43)(36,44)(45,47)(46,48);(1,3,5,11)(2,6)(4,10,13,16)(7,14)(8,9)(12,15)(17,19)(18,23)(20,30,33,28)(21,35)(22,31)(24,41,26,38)(25,43)(27,40)(29,37)(32,46,36,45)(34,39)(42,48,44,47);(1,4,5,13)(2,7)(3,10)(6,14,9,15)(8,12)(11,16)(17,20,22,33)(18,24,27,26)(19,28,31,30)(21,32)(23,38,40,41)(25,42)(29,45)(34,44)(35,46)(36,37)(39,47)(43,48);(2,8)(14,15)(17,21,22,37)(18,25,27,34)(19,29,31,35)(20,32,33,36)(23,39,40,43)(24,42,26,44)(28,45,30,46)(38,47,41,48);(3,11)(4,13)(6,9)(10,16)(17,20)(18,26)(19,30)(21,36)(22,33)(23,38)(24,27)(25,42)(28,31)(29,45)(32,37)(34,44)(35,46)(39,48)(40,41)(43,47)

Polycyclic presentation:

2^10|b^2=f,c^2=fgj,d^2=j,b^a=bg,c^a=cfgh,c^b=cfghi,d^a=dfhij,d^b=dfij,d^c=dfij,e^a=efghj,e^b=efghij,e^c=efgj,e^d=ej,f^a=fi,f^c=fj,f^e=fj,g^b=gi,g^c=gj,g^d=gj,g^e=gj,h^a=hi,h^b=hij,h^c=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341033, #72341097, #72341103
maximal normal subgroup of	#72341033, #72341097, #72341103

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