#72285392

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5768 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fg,e^2=j,b^a=bg,c^a=cfgh,c^b=cfgh,d^a=dhi,d^b=dfgh,d^c=dfgij,e^a=eij,e^b=efhij,e^c=efhi,f^b=fi,f^c=fij,f^d=fj,g^a=gi,g^c=gij,g^d=gj,g^e=gj,h^a=hj,h^b=hi,h^c=hi

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285599
maximal normal subgroup of	#72285599

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