#72340479

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
prime radical	#245249	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2782 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)(4,8)(5,11)(6,10)(9,15)(12,14)(13,16)(20,31)(21,33)(23,27)(47,48);(1,3)(2,7)(4,11,13,9)(5,8,15,16)(6,12)(10,14)(17,18)(19,27,30,23)(20,24,31,40)(21,34,33,25)(22,26)(28,38)(29,42)(32,39)(35,41)(36,44)(37,46)(43,47,45,48);(1,4)(2,8,10,16)(3,11,12,9)(5,14)(6,13)(7,15)(17,19)(18,23)(20,28,31,44)(21,29)(22,30)(24,38,40,36)(25,37)(26,27)(32,43,35,45)(33,46)(34,42)(39,48,41,47);(1,5,6,15)(2,9)(3,8,12,16)(4,7)(10,11)(13,14)(17,20)(18,24,26,40)(19,28)(21,35,33,32)(22,31)(23,36,27,38)(25,41)(29,45,46,43)(30,44)(34,39)(37,48)(42,47);(4,13)(7,14)(8,16)(9,11)(17,21)(18,25)(19,29)(20,32)(22,33)(23,37)(24,39)(26,34)(27,42)(28,43)(30,46)(31,35)(36,47)(38,48)(40,41)(44,45)

Polycyclic presentation:

2^10|b^2=f,c^2=gh,d^2=fh,b^a=bf,c^a=cfg,c^b=cfgij,d^a=dgi,d^b=dfg,d^c=dgi,e^a=egi,e^b=ehij,e^c=eij,e^d=egh,f^c=fi,f^d=fi,f^e=fj,g^b=gi,g^c=gj,g^d=gij,h^a=hij,h^b=hj,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340480, #72340498
maximal normal subgroup of	#72340480, #72340498

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