#72307205

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#23960
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#364376
prime radical	#7622837	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3717 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=fghj,d^2=ij,g^2=ij,h^2=j,i^2=j,b^a=bf,c^a=cg,c^b=cghj,d^a=dij,d^b=dfhj,d^c=dfhi,e^d=ej,f^c=fj,f^d=fj,g^a=gi,g^b=gij,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72320238, #72321054
maximal normal subgroup of	#72307301, #72320238, #72321054

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