#72287693

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21690
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#33748	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #7846 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

923 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=gj,c^2=e,d^2=ei,e^2=i,f^2=hj,g^2=ij,h^2=j,b^a=bg,c^a=cf,c^b=cgij,d^a=dhj,d^b=dij,e^a=ej,f^a=fh,f^c=fhj,f^d=fj,g^a=gij,g^c=gij,h^a=hj,h^c=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287720
maximal normal subgroup of	#72287720

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