#72310859

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

2878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=g,b^a=bf,c^a=ch,c^b=ci,d^c=dg,e^a=ehi,e^b=ehi,e^c=ei,e^d=efg,f^b=fi,f^c=fj,f^e=fj,g^a=gi,g^e=gj,h^b=hj,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310874, #72340239, #72340307, …
maximal normal subgroup of	#72310874, #72340239, #72340307, …

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