#72303135

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#417526
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#7622875	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

4682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=fj,d^2=egi,b^a=bh,c^b=cf,d^a=dg,d^b=de,d^c=defgj,e^a=ej,e^b=ei,f^d=fj,g^b=gij,g^c=gj,h^d=hi,h^e=hj,h^g=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303157, #72303348, #72303445
maximal normal subgroup of	#72303157, #72303348, #72303445

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