#72284527

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	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233559	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2574 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)(3,13,17,8)(4,9)(5,10,26,37)(6,11)(7,12)(14,31,43,28)(15,29,47,58)(16,30,18,41)(19,49,44,57)(20,51,53,61)(21,33)(22,34)(23,35)(24,32,54,62)(25,38,52,36)(27,39)(40,46,59,48)(42,60,50,63)(45,56,55,64);(1,3)(2,8)(4,19)(5,15)(6,18)(7,17)(9,31)(10,29)(11,30)(12,13)(14,22)(16,27)(20,50)(21,43)(23,44)(24,55)(25,48)(26,46)(28,35)(32,60)(33,57)(34,49)(36,58)(37,59)(38,40)(39,41)(42,54)(45,53)(47,52)(51,56)(61,63)(62,64);(1,4)(2,9)(3,14,17,43)(5,24,26,54)(6,21)(7,23)(8,28,13,31)(10,32,37,62)(11,33)(12,35)(15,45,47,55)(16,19,18,44)(20,52,53,25)(22,27)(29,56,58,64)(30,49,41,57)(34,39)(36,61,38,51)(40,60,59,63)(42,46,50,48);(1,5,6,26)(2,10)(3,15,18,46)(4,20,22,54)(7,25,27,52)(8,29)(9,32,35,51)(11,37)(12,38)(13,40)(14,42,19,50)(16,47,17,48)(21,53,23,24)(28,56,31,60)(30,59)(33,62,34,61)(36,39)(41,58)(43,45,44,55)(49,63,57,64);(2,11)(3,16)(5,25)(8,13)(9,33)(10,36)(12,39)(14,19)(15,46)(17,18)(20,24)(26,52)(28,31)(30,41)(32,61)(34,35)(37,38)(42,55)(43,44)(45,50)(47,48)(49,57)(51,62)(53,54)

Polycyclic presentation:

2^10|a^2=f,c^2=f,d^2=h,b^a=bg,c^b=ch,d^a=dfhi,d^c=dgi,e^a=ei,e^b=eij,e^d=ej,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^a=hi,h^c=hj

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709
maximal normal subgroup of	#72284709

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