#72340455

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

2835 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)(3,11)(4,13)(5,9)(6,10)(7,12)(8,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,40)(28,39)(29,35)(30,38)(31,45)(32,36)(33,42)(34,37)(41,46)(43,47)(44,48);(1,3,6,12)(2,7,10,11)(4,9,14,16)(5,13,15,8)(18,23)(24,35)(25,36)(26,37)(27,30)(28,33)(31,46)(43,44);(1,4,6,14)(2,8)(3,9,12,16)(5,11)(7,15)(10,13)(17,19)(18,24)(20,30,32,27)(21,33,34,28)(22,29)(23,35)(25,40,36,38)(26,42,37,39)(31,43)(41,47)(44,46)(45,48);(1,5,6,15)(2,9)(3,8)(4,11,14,7)(10,16)(12,13)(17,20)(18,25)(19,27,29,30)(21,31)(22,32)(23,36)(24,38,35,40)(26,41)(28,43,33,44)(34,46)(37,45)(39,47,42,48);(2,10)(3,12)(5,15)(7,11)(17,21)(18,26)(19,28)(20,31)(22,34)(23,37)(24,39)(25,41)(27,43)(29,33)(30,44)(32,46)(35,42)(36,45)(38,47)(40,48)

Polycyclic presentation:

2^10|b^2=i,c^2=fi,d^2=gh,b^a=bij,c^a=ch,c^b=cf,d^a=dfg,d^b=dfgi,d^c=dgj,e^a=ehi,e^b=eg,e^c=eghi,e^d=ei,f^a=fi,f^d=fij,f^e=fj,g^a=gi,g^c=gij,g^d=gj,h^b=hi,h^c=hi,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340457, #72340486, #72340498, …
maximal normal subgroup of	#72340457, #72340486, #72340498, …

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