#72340317

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

2789 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)(3,7,11,5)(4,12,14,8)(6,9)(10,15)(13,16)(17,18)(19,23,29,38)(20,30,33,24)(21,25,35,42)(22,26)(27,43)(28,40)(31,36)(32,39)(34,37)(41,46)(44,48,45,47);(1,3)(2,7,9,5)(4,10)(6,11)(8,15,12,13)(14,16)(17,19)(18,23,26,38)(20,31)(21,34)(22,29)(24,36,30,43)(25,40,42,37)(27,33)(28,35)(32,44)(39,48,41,47)(45,46);(1,4,6,14)(2,8,9,12)(3,10,11,16)(5,13,7,15)(21,35)(28,34)(32,46)(44,45);(1,5,6,7)(2,3)(4,13,14,15)(8,10)(9,11)(12,16)(17,20)(18,24)(19,27)(21,32,35,46)(22,33)(23,36)(25,41,42,39)(26,30)(28,45,34,44)(29,31)(37,47,40,48)(38,43);(8,12)(13,15)(17,21)(18,25)(19,28)(20,32)(22,35)(23,37)(24,39)(26,42)(27,44)(29,34)(30,41)(31,45)(33,46)(36,47)(38,40)(43,48)

Polycyclic presentation:

2^10|a^2=f,b^2=fg,c^2=ij,d^2=ghij,b^a=bg,c^a=cgh,d^a=dfg,d^b=dfhj,e^a=ehj,e^b=ej,e^c=efgj,e^d=ehj,f^b=fi,f^c=fij,f^d=fi,f^e=fj,g^a=gi,g^b=gi,g^c=gij,g^d=gj,g^e=gj,h^a=hj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340328, #72340368
maximal normal subgroup of	#72340328, #72340368

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