#72341085

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
prime radical	#245255	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2773 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)(3,7,5,11)(4,12)(6,9)(8,14)(10,15,13,16)(17,18,22,26)(19,23,29,36)(20,24,32,38)(21,25)(27,34,30,43)(28,41)(31,45)(33,39)(35,42)(37,46)(40,47)(44,48);(1,3)(2,7,9,11)(4,13)(5,6)(8,15,12,16)(10,14)(17,19)(18,23)(20,30,32,27)(21,28)(22,29)(24,34,38,43)(25,35)(26,36)(31,40,46,44)(33,42)(37,48,45,47)(39,41);(1,4,6,14)(2,8,9,12)(3,10)(5,13)(7,15)(11,16)(17,20)(18,24)(19,27)(21,31,33,46)(22,32)(23,34)(25,37,39,45)(26,38)(28,40,42,44)(29,30)(35,47,41,48)(36,43);(1,2)(3,11,5,7)(4,12,14,8)(6,9)(10,16)(13,15)(17,21,22,33)(18,25)(19,28)(20,31)(23,35,36,41)(24,37,38,45)(26,39)(27,40,30,44)(29,42)(32,46)(34,47)(43,48);(1,5,6,3)(2,7,9,11)(4,10,14,13)(8,15,12,16)(20,32)(34,43)(37,45)(40,44)

Polycyclic presentation:

2^10|a^2=f,b^2=gj,c^2=fij,d^2=fghj,e^2=i,b^a=bfg,c^a=cfij,c^b=ch,d^a=dghj,d^b=dfgij,d^c=dfg,e^a=egj,e^b=egij,e^c=ehj,e^d=egij,f^b=fi,f^d=fj,f^e=fi,g^a=gi,g^c=gj,g^d=gi,h^a=hij,h^b=hj,h^c=hi,h^d=hij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341091, #72341110, #72341114
maximal normal subgroup of	#72341091, #72341110, #72341114

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