#72340489

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	#251860	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2710 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)(4,8)(5,11)(6,10)(9,15)(12,14)(13,16)(17,18)(19,23)(20,30)(21,34)(22,26)(24,33)(25,36)(27,40)(28,43)(29,39)(31,37)(32,44)(35,38)(41,46)(42,48)(45,47);(1,3,6,12)(2,7,10,14)(4,9)(5,8)(11,13)(15,16)(17,19,22,29)(18,23,26,39)(20,31,33,27)(21,35)(24,40,30,37)(25,38)(28,36)(32,42)(34,43)(41,48)(44,47)(45,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(24,30)(25,34)(27,31)(28,35)(32,46)(41,44)(42,45)(47,48);(1,5,6,15)(2,9)(3,8,12,16)(4,7)(10,11)(13,14)(17,20)(18,24,26,30)(19,27,29,31)(21,32)(22,33)(23,37)(25,41,34,44)(28,42,35,45)(36,46)(38,47)(39,40)(43,48);(3,12)(4,13)(5,15)(9,11)(17,21)(18,25)(19,28)(20,32)(22,36)(23,38)(24,41)(26,34)(27,42)(29,35)(30,44)(31,45)(33,46)(37,47)(39,43)(40,48)

Polycyclic presentation:

2^10|b^2=fij,d^2=gi,b^a=bf,c^a=cfh,c^b=cfh,d^a=dfh,d^b=dhi,d^c=dfg,e^a=ehj,e^b=egi,e^c=eghi,e^d=ei,f^c=fi,f^d=fi,f^e=fj,g^a=gij,g^b=gj,g^c=gi,h^b=hi,h^c=hi,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340495
maximal normal subgroup of	#72340495

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