#72318615

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32898
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#8926850	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3978 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|c^2=hij,g^2=j,b^a=bhj,c^a=ce,c^b=cef,d^c=dg,e^b=eh,e^d=ei,f^a=fh,f^b=fhj,f^c=fj,f^d=fi,f^e=fj,g^a=gij,g^c=gj,g^d=gj,g^e=gj,g^f=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318703, #72318857, #72319043, …
maximal normal subgroup of	#72318703, #72318857, #72319043, …

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