#72310996

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

2857 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=i,e^2=gi,c^a=cg,c^b=cf,d^a=dh,d^b=dhi,d^c=dhi,e^b=ehi,e^d=efg,f^a=fi,f^c=fi,f^d=fj,f^e=fj,g^b=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340222, #72340228, #72340254, …
maximal normal subgroup of	#72340222, #72340228, #72340254, …

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