#72307102

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24196
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24196
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,9)(4,18,21,45)(5,23)(6,26)(7,8)(10,22,29,52)(11,25)(12,24)(13,28)(14,38,20,47)(15,34)(16,43)(17,32)(19,46,49,48)(27,55)(30,56)(31,41,36,51)(33,44)(35,53,54,50)(37,60)(39,61,40,62)(42,57)(58,64,59,63);(2,8)(9,28)(10,29)(11,23)(12,30)(18,45)(26,55)(31,36)(32,34)(33,57)(35,54)(38,47)(43,60)(46,48)(58,59)(61,62);(3,13)(4,19,21,49)(6,27)(9,28)(10,35,29,54)(12,30)(14,39,20,40)(15,17)(18,46,45,48)(22,53,52,50)(24,56)(26,55)(31,58,36,59)(32,34)(38,61,47,62)(41,64,51,63);(1,3,7,13)(2,9,8,28)(4,20)(5,17,25,15)(6,16,27,37)(10,36)(11,34,23,32)(12,33,30,57)(14,21)(18,47)(19,40)(22,51)(24,44,56,42)(26,43,55,60)(29,31)(35,59)(38,45)(39,49)(41,52)(46,62)(48,61)(50,64)(53,63)(54,58);(1,4,6,22,5,19,24,50,7,21,27,52,25,49,56,53)(2,10,12,18,11,35,26,48,8,29,30,45,23,54,55,46)(3,14,16,41,17,39,44,63,13,20,37,51,15,40,42,64)(9,31,33,38,34,58,43,62,28,36,57,47,32,59,60,61)

Polycyclic presentation:

2^10|a^2=g,c^2=g,d^2=gj,e^2=h,f^2=j,h^2=i,i^2=j,b^a=bj,d^c=dgj,e^a=eh,e^c=ef,e^d=efij,f^a=fj,f^d=fj,g^e=gj,h^a=hi,h^c=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307129, C₂×#72307102, #72307810
maximal normal subgroup of	#72307129, C₂×#72307102, #72307810
maximal quotient of	C₂×#72307102

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