#72307253

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	#21529
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#365008
prime radical	#7622966	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2379 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:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=e,d^2=eghj,e^2=i,g^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cg,d^a=df,d^b=dfij,d^c=dfi,e^b=ei,e^d=ej,f^b=fj,g^b=gi,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72343946, #72343971
maximal normal subgroup of	#72307299, #72343946, #72343971

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