#72341087

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	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2904 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=gh,c^2=h,d^2=fhij,e^2=ij,b^a=bfg,c^a=cfhj,c^b=cfj,d^a=dfgij,d^b=dgij,d^c=dghij,e^a=eghij,e^b=eghij,e^c=ehi,e^d=egij,f^b=fj,f^c=fi,f^d=fj,f^e=fij,g^a=gj,g^b=gi,g^c=gj,g^d=gij,h^a=hi,h^b=hi,h^d=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341091, #72341103, #72341124
maximal normal subgroup of	#72341091, #72341103, #72341124

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