#72340500

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

2847 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18)(19,27,28,23)(20,31,33,24)(21,25,35,41)(22,26)(29,38)(30,37)(32,36)(34,44)(39,46)(40,42)(43,47,45,48);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,19)(18,23,26,27)(20,32)(21,30,35,42)(22,28)(24,38,31,36)(25,40)(29,33)(34,45,46,43)(37,41)(39,47)(44,48);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(19,28)(20,33)(23,27)(30,42)(36,38)(37,40)(39,44)(43,45);(2,8)(6,14)(17,20)(18,24)(19,29)(21,34)(22,33)(23,36)(25,39)(26,31)(27,38)(28,32)(30,43)(35,46)(37,47)(40,48)(41,44)(42,45);(2,8)(4,12)(6,14)(7,11)(17,21)(18,25)(19,30)(20,34)(22,35)(23,37)(24,39)(26,41)(27,40)(28,42)(29,43)(31,44)(32,45)(33,46)(36,47)(38,48)

Polycyclic presentation:

2^10|a^2=f,b^2=fgj,c^2=ij,b^a=bg,c^a=cgh,c^b=cj,d^a=dhi,d^b=dj,d^c=dfg,e^a=ehij,e^b=eh,e^c=eghij,f^b=fi,f^c=fij,f^d=fj,f^e=fj,g^a=gi,g^b=gi,g^c=gij,g^d=gj,h^a=hj,h^b=hj,h^c=hij

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340502
maximal normal subgroup of	#72340502

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