#72307236

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#24089
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#368007
prime radical	#60450	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2393 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fj,c^2=efgj,d^2=i,g^2=i,h^2=j,i^2=j,b^a=bf,c^a=ch,c^b=cg,d^a=dfhi,d^b=dfh,d^c=dfhi,e^c=ej,f^c=fj,f^d=fj,g^a=gj,g^b=gi,h^d=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72320114, #72320989
maximal normal subgroup of	#72307301, #72320114, #72320989

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