#72284747

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	#251859	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2747 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:

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

Polycyclic presentation:

2^10|c^2=fh,d^2=gh,e^2=ghij,c^b=cf,d^a=dfhi,d^b=dfg,d^c=dhj,e^a=ef,e^b=ef,e^c=egij,e^d=eij,f^c=fi,f^d=fij,f^e=fi,g^a=gi,g^c=gij,g^d=gj,g^e=gj,h^b=hi,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599, #72285772, #72340222, …
maximal normal subgroup of	#72285599, #72285772, #72340222, …

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