#72340346

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	#251855	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2731 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)(3,6)(4,11,12,7)(5,8)(9,14,16,15)(10,13)(17,18)(19,27,30,23)(20,31,34,24)(21,25,35,40)(22,26)(28,38)(29,42)(32,36)(33,39)(37,44)(41,46)(43,47,45,48);(4,12)(6,13)(7,11)(9,16)(17,19)(18,23)(20,32)(21,29)(22,30)(24,38)(25,37)(26,27)(28,34)(31,36)(33,45)(35,44)(39,48)(40,42)(41,47)(43,46);(1,3,5,10)(2,6,8,13)(4,9,12,16)(7,15,11,14)(21,35)(25,40)(43,45)(47,48);(1,4)(2,7,8,11)(3,9)(5,12)(6,14,13,15)(10,16)(17,20,22,34)(18,24,26,31)(19,28,30,32)(21,33)(23,36,27,38)(25,41)(29,43)(35,46)(37,48)(39,40)(42,47)(44,45);(3,10)(14,15)(17,21)(18,25)(19,29)(20,33)(22,35)(23,37)(24,39)(26,40)(27,42)(28,43)(30,44)(31,41)(32,45)(34,46)(36,47)(38,48)

Polycyclic presentation:

2^10|a^2=f,c^2=ij,d^2=gj,b^a=bh,c^b=cg,d^a=dfghij,d^b=di,d^c=dg,e^a=ehj,e^c=efgi,e^d=ehj,f^b=fj,f^d=fi,f^e=fj,g^a=gij,g^e=gj,h^a=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340368, #72340470
maximal normal subgroup of	#72340368, #72340470

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