#72282014

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3155	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2960 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2)(3,4)(5,6)(7,14)(8,15)(9,29)(10,35)(11,39)(12,40)(13,19)(16,31)(17,37)(18,38)(20,36)(21,55)(22,48)(23,47)(24,49)(25,59)(26,44)(27,43)(28,45)(30,65)(32,53)(33,63)(34,51)(41,64)(42,56)(46,60)(50,61)(52,66)(54,67)(57,68)(58,62);(5,7)(6,14)(8,13)(9,25,33,21)(10,22)(11,23)(12,24)(15,19)(16,42,52,46)(17,43)(18,44)(20,45)(26,38)(27,37)(28,36)(29,55,63,59)(30,41,58,34)(31,60,66,56)(32,61,67,57)(35,48)(39,47)(40,49)(50,53,68,54)(51,62,64,65);(1,2)(3,4)(5,8,13,7)(6,15,19,14)(9,30)(10,26,38,22)(11,27,37,23)(12,28,36,24)(16,50)(17,47,39,43)(18,48,35,44)(20,49,40,45)(21,34)(25,41)(29,62)(31,57)(32,60)(33,58)(42,53)(46,54)(51,59)(52,68)(55,64)(56,67)(61,66)(63,65);(2,4)(5,9,12,41,13,33,36,34)(6,16,20,54,19,52,40,53)(7,21,24,30,8,25,28,58)(10,32,37,66,38,67,11,31)(14,42,45,50,15,46,49,68)(17,51,35,63,39,64,18,29)(22,57,27,60,26,61,23,56)(43,62,48,59,47,65,44,55)

Polycyclic presentation:

2^10|b^2=h,c^2=hj,d^2=hi,e^2=ij,f^2=i,i^2=j,b^a=bh,c^a=ch,c^b=chj,d^a=de,d^b=dg,d^c=defij,e^a=ei,e^d=eij,f^a=fij,f^d=fi,g^a=gj,g^b=gj,g^c=gj,h^d=hj,i^a=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282392
maximal normal subgroup of	#72282392

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