#72306307

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5625 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=hij,e^2=fghj,f^2=j,g^2=i,h^2=ij,b^a=bf,c^a=ch,c^b=cgh,d^a=dj,d^c=di,e^a=ehj,e^b=eij,e^c=efgj,e^d=ei,f^a=fj,f^b=fi,f^c=fi,f^e=fi,g^a=gij,g^c=gij,h^a=hij,h^b=hi,h^e=hi

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306341
maximal normal subgroup of	#72306341

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