#72306408

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	D₄×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5968 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)(8,14)(9,12)(15,16)(17,18)(19,24)(20,25)(21,33)(22,26)(23,27)(28,40)(29,37)(30,38)(31,39)(32,41)(34,43)(35,42)(36,44)(45,48)(46,47);(2,8)(3,6)(4,11)(7,14)(9,12)(10,13)(17,19,23,31)(18,24,27,39)(20,29,21,28)(22,35,36,30)(25,40,33,37)(26,42,44,38)(32,45,34,46)(41,47,43,48);(1,2)(3,8)(4,12)(5,7)(6,14)(9,13)(10,16)(11,15)(17,20,23,21)(18,25,27,33)(19,28)(22,34,36,32)(24,37)(26,43,44,41)(29,31)(30,46)(35,45)(38,48)(39,40)(42,47);(1,3)(2,7)(5,6)(9,16)(10,11)(12,15)(17,21,23,20)(18,25,27,33)(19,29,31,28)(22,34,36,32)(24,37,39,40)(26,41,44,43)(30,46,35,45)(38,47,42,48);(1,4,5,13)(2,9,7,12)(3,10,6,11)(8,15,14,16)(17,22,23,36)(18,26,27,44)(19,30,31,35)(20,32,21,34)(24,38,39,42)(25,41,33,43)(28,45,29,46)(37,47,40,48)

Polycyclic presentation:

2^10|b^2=i,c^2=fgij,d^2=i,e^2=ij,h^2=j,b^a=bf,c^a=cfg,c^b=ch,d^a=dfgij,d^b=dij,d^c=dgh,e^b=efhij,e^c=eij,e^d=efhj,f^c=fi,f^d=fi,g^b=gi,g^c=gi,g^d=gi,h^b=hj,h^c=hij,h^d=hij,h^e=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306523
maximal normal subgroup of	#72306523

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