#72307202

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21526
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#354595
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20331 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2241 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=ef,d^2=efgi,g^2=i,h^2=j,i^2=j,b^a=bh,d^a=dfgij,d^b=dfgj,d^c=dij,e^a=ej,f^a=fj,f^d=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,h^a=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72319811, #72320921
maximal normal subgroup of	#72307299, #72319811, #72320921

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