#72341099

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	#35527	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2701 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)(3,7,10,14)(4,11)(5,6)(8,13)(9,15,12,16)(17,18)(19,23)(20,30,33,24)(21,26)(22,25)(27,36,31,39)(28,38)(29,37)(32,43,34,40)(35,41)(42,45)(44,47,46,48);(1,3)(2,7,5,14)(4,12)(6,10)(8,15,11,16)(9,13)(17,19,22,28)(18,23,25,38)(20,31,33,27)(21,29)(24,39,30,36)(26,42)(32,44)(34,46)(35,45)(37,41)(40,48)(43,47);(1,4)(2,8,5,11)(3,9)(6,13)(7,15,14,16)(10,12)(17,20,22,33)(18,24,25,30)(19,27,28,31)(21,34)(23,36,38,39)(26,43)(29,46)(32,35)(37,48)(40,41)(42,47)(44,45);(1,5,6,2)(3,7,10,14)(4,8,13,11)(9,15,12,16)(18,25)(19,28)(26,41)(29,45);(3,10)(7,14)(17,21,22,35)(18,26,25,41)(19,29,28,45)(20,32,33,34)(23,37,38,42)(24,40,30,43)(27,44,31,46)(36,47,39,48)

Polycyclic presentation:

2^10|a^2=g,b^2=fgi,c^2=fgi,d^2=j,e^2=i,b^a=bf,c^a=ch,c^b=cfgij,d^a=dgij,d^b=dgi,d^c=dh,e^b=eghij,e^c=eg,f^a=fj,f^b=fj,f^c=fi,f^d=fj,f^e=fi,g^b=gj,g^c=gi,h^a=hi,h^b=hij,h^c=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341103, #72341114
maximal normal subgroup of	#72341103, #72341114

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