#72310961

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

2963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|c^2=g,d^2=gj,b^a=bij,c^a=ch,c^b=ch,d^a=df,d^b=dfg,d^c=dgi,e^a=eh,e^c=efgh,e^d=eh,f^b=fi,f^c=fij,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^e=gj,h^c=hi,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311029, #72340222, #72340284, …
maximal normal subgroup of	#72311029, #72340222, #72340284, …

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