#72311397

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

2850 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)(4,12,14,8)(5,9)(7,11)(10,15,13,16)(17,18)(19,27,30,24)(20,25,33,35)(21,26)(22,23)(28,41)(29,42,44,38)(31,37)(32,39,46,40)(34,36)(43,48)(45,47);(1,3)(2,6)(4,10)(5,11)(7,9)(8,16)(12,15)(13,14)(18,23)(21,34)(24,27)(25,35)(28,31)(38,42)(39,40)(47,48);(1,3)(2,7,9,6)(4,13,14,10)(5,11)(8,16)(12,15)(17,19)(18,24)(20,31,33,28)(21,29,34,44)(22,30)(23,27)(25,37,35,41)(26,38,36,42)(32,43)(39,48)(40,47)(45,46);(1,4)(2,8,9,12)(3,10)(5,14)(6,15,7,16)(11,13)(17,20)(18,25)(19,28,30,31)(21,32,34,46)(22,33)(23,35)(24,37,27,41)(26,40,36,39)(29,43)(38,48)(42,47)(44,45);(2,9)(4,14)(6,7)(15,16)(17,21,22,34)(18,26,23,36)(19,29,30,44)(20,32,33,46)(24,38,27,42)(25,39,35,40)(28,43,31,45)(37,47,41,48)

Polycyclic presentation:

2^10|a^2=f,c^2=fg,d^2=g,e^2=j,b^a=bj,c^a=ch,c^b=cfg,d^a=dfghi,d^b=dg,d^c=dghj,e^a=eij,e^b=efj,e^c=efj,e^d=ehi,f^c=fj,f^d=fi,g^a=gij,g^c=gj,g^e=gj,h^a=hj,h^b=hij,h^c=hij,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340222, #72340228, #72340246, …
maximal normal subgroup of	#72340222, #72340228, #72340246, …

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