#72227965

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁷	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

10238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gj,c^2=ghi,d^2=j,e^2=gh,f^2=gij,b^a=bh,c^a=ch,c^b=ci,d^a=dhi,d^b=di,d^c=dgi,e^a=eghj,e^b=eghj,e^c=ej,e^d=eg,f^a=fhj,f^b=fgh,f^c=fgh,f^d=fhij,f^e=fghij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227966
Fitting subgroup of	#72227966
maximal subgroup of	#72227966
maximal normal subgroup of	#72227966

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