#72307179

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	#24338
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#368117
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2398 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=fj,c^2=ei,d^2=efgj,g^2=i,i^2=j,b^a=bf,c^a=cfh,c^b=cfhi,d^a=dhij,d^b=dg,d^c=dfhij,e^d=ej,f^c=fj,f^d=fj,g^a=gj,g^b=gi,g^c=gj,h^b=hj,h^c=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307301, #72319909, #72320733
maximal normal subgroup of	#72307301, #72319909, #72320733

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