#72285290

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

2845 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)(4,13)(5,8)(6,9)(10,12)(11,14)(15,16)(19,27)(20,31)(24,40)(29,32)(30,35)(33,36)(41,46)(44,45);(1,3)(2,7)(4,10,8,15)(5,12,13,16)(6,11)(9,14)(17,18)(19,28,27,23)(20,24)(21,34)(22,26)(25,37)(29,43,32,38)(30,39,35,42)(31,40)(33,41)(36,46)(44,48,45,47);(1,4)(2,5,9,13)(3,10,11,15)(6,8)(7,12)(14,16)(17,19)(18,23)(20,32,31,29)(21,35,37,30)(22,27)(24,38,40,43)(25,42,34,39)(26,28)(33,45)(36,44)(41,47)(46,48);(3,11)(4,8)(7,14)(12,16)(17,20)(18,24)(19,29)(21,36)(22,31)(23,38)(25,41)(26,40)(27,32)(28,43)(30,45)(33,37)(34,46)(35,44)(39,47)(42,48);(1,5)(2,8)(3,12,11,16)(4,9)(6,13)(7,10,14,15)(17,21)(18,25,26,34)(19,30,27,35)(20,33,31,36)(22,37)(23,39)(24,41)(28,42)(29,44)(32,45)(38,47,43,48)(40,46)

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,e^2=gj,b^a=bf,c^a=cfg,c^b=cfgj,d^a=dfj,d^b=dij,d^c=dfhi,e^b=efgh,e^c=eghi,e^d=ehj,f^c=fi,f^e=fij,g^b=gi,g^c=gj,g^d=gj,h^a=hij,h^b=hj,h^c=hij,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526
maximal normal subgroup of	#72285526

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