#72340306

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

2768 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)(4,12)(5,8)(7,14)(9,11)(10,16)(13,15)(17,18)(19,23)(20,30)(21,25)(22,26)(24,33)(27,36)(28,41)(29,35)(31,43)(32,38)(34,39)(37,42)(40,47)(44,46)(45,48);(1,2)(3,9,11,6)(4,7,14,12)(5,8)(10,16)(13,15)(17,19,22,29)(18,23,26,35)(20,31,33,27)(21,28)(24,36,30,43)(25,37)(32,40)(34,42)(38,48)(39,41)(44,47)(45,46);(1,3,5,11)(2,6,8,9)(4,13,14,10)(7,15,12,16)(21,34)(23,35)(24,30)(40,45);(1,4)(2,7,8,12)(3,10,11,13)(5,14)(6,15)(9,16)(17,20,22,33)(18,24)(19,27,29,31)(21,32,34,46)(23,36)(25,38)(26,30)(28,40,42,45)(35,43)(37,47)(39,44)(41,48);(4,14)(6,9)(7,12)(10,13)(17,21)(18,25)(19,28)(20,32)(22,34)(23,37)(24,38)(26,39)(27,40)(29,42)(30,44)(31,45)(33,46)(35,41)(36,47)(43,48)

Polycyclic presentation:

2^10|b^2=fj,c^2=i,d^2=gj,b^a=bf,c^a=cfh,c^b=cfg,d^a=dfh,d^b=dfgj,d^c=dfg,e^a=eh,e^b=ehj,e^c=eghj,e^d=ei,f^c=fi,f^d=fi,f^e=fj,g^a=gij,g^b=gi,g^c=gi,h^b=hj,h^c=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340328
maximal normal subgroup of	#72340328

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