#72228716

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄

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

10070 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,c^2=gij,d^2=ghi,e^2=ij,f^2=ghij,b^a=bi,c^a=ci,c^b=ch,d^a=dj,d^b=dh,d^c=dg,e^a=egh,e^b=egij,e^c=ei,e^d=egh,f^b=fgj,f^c=fghi,f^d=fh,f^e=fghij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72228730
Fitting subgroup of	#72228730
maximal subgroup of	#72228730
maximal normal subgroup of	#72228730

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