#72317626

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9497
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
socle	C₂	#119494
prime radical	#62811	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

2614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=hj,e^2=j,b^a=bf,c^a=cej,c^b=cd,d^a=dh,f^a=fi,f^c=fh,g^b=gi,g^c=gj,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317635, #72317758, #72317779
maximal normal subgroup of	#72317635, #72317758, #72317779

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