#72310958

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2859 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)(3,9)(4,12,15,7)(5,11)(6,8)(10,14,13,16)(17,18)(19,23)(20,24,31,37)(21,25)(22,26)(27,34)(28,35,41,45)(29,36)(30,38,33,39)(32,40)(42,46)(43,47,44,48);(1,3)(2,5)(4,13)(6,11)(7,16)(8,9)(10,15)(12,14)(17,19)(18,23)(20,28)(21,29)(22,27)(24,35)(25,36)(26,34)(30,44)(31,41)(32,42)(33,43)(37,45)(38,48)(39,47)(40,46);(1,4)(2,7)(3,10)(5,14)(6,15)(8,12)(9,16)(11,13)(19,27)(21,32)(23,34)(24,37)(28,41)(36,46)(43,44)(47,48);(1,5)(2,3,8,11)(4,14)(6,9)(7,13,12,10)(15,16)(17,20)(18,24,26,37)(19,28)(21,33,32,30)(22,31)(23,35,34,45)(25,39)(27,41)(29,44,42,43)(36,48)(38,40)(46,47);(3,11)(5,9)(17,21)(18,25)(19,29)(20,30)(22,32)(23,36)(24,38)(26,40)(27,42)(28,43)(31,33)(34,46)(35,47)(37,39)(41,44)(45,48)

Polycyclic presentation:

2^10|a^2=h,d^2=g,b^a=bi,c^a=cf,c^b=cij,d^a=dfg,d^b=dfh,d^c=dg,e^b=ehi,e^c=efgj,e^d=ehi,f^a=fi,f^d=fij,f^e=fj,g^a=gij,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	#72310982, #72340222, #72340258, …
maximal normal subgroup of	#72310982, #72340222, #72340258, …

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