#72340490

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	#245231	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2733 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)(4,8)(5,11)(6,10)(9,15)(12,14)(13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,33)(27,40)(28,38)(29,39)(31,37)(32,41)(34,46)(35,43)(36,42)(44,47)(45,48);(1,3,6,12)(2,7,10,14)(4,9)(5,8)(11,13)(15,16)(17,19,22,29)(18,23,26,39)(20,31,33,27)(21,34)(24,40,30,37)(25,38)(28,35)(32,44)(36,45)(41,48)(42,47)(43,46);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(21,35)(24,30)(27,31)(32,36)(38,46)(41,42)(44,45)(47,48);(1,5,6,15)(2,9)(3,8,12,16)(4,7)(10,11)(13,14)(17,20)(18,24,26,30)(19,27,29,31)(21,36,35,32)(22,33)(23,37)(25,42)(28,45)(34,44)(38,48,46,47)(39,40)(41,43);(2,10)(4,13)(8,16)(9,11)(17,21,22,35)(18,25,26,43)(19,28,29,34)(20,32,33,36)(23,38,39,46)(24,41,30,42)(27,44,31,45)(37,47,40,48)

Polycyclic presentation:

2^10|b^2=fij,d^2=gi,e^2=j,b^a=bf,c^a=cfh,c^b=cfh,d^a=dfh,d^b=dhi,d^c=dfg,e^a=ehi,e^b=efg,e^c=efgij,e^d=eh,f^c=fi,f^d=fi,f^e=fj,g^a=gij,g^b=gj,g^c=gi,g^e=gj,h^b=hi,h^c=hi,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340495
maximal normal subgroup of	#72340495

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