#72307101

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	#24089
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂	C₂×#24089
prime radical	#10576609	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4752 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=hj,d^2=g,f^2=j,g^2=i,i^2=j,b^a=bj,c^a=chj,d^a=dg,d^c=df,e^c=ehj,f^a=fj,f^d=fj,f^e=fj,g^a=gi,h^d=hj,i^a=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307130, #72307386
maximal normal subgroup of	#72307130, #72307386

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