#72339880

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#499831
commutator, Frattini, cosocle	C₂×#127	C₂⁴
LCS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #117 ◁ #3508 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,16)(5,11)(6,9)(8,18)(10,20)(12,28)(13,27)(14,22)(15,23)(17,26)(19,25)(21,29)(24,31)(30,32);(1,3)(2,7)(4,12,17,29,19,30,18,31)(5,13)(6,15)(8,24,16,28,26,21,25,32)(9,23)(10,22)(11,27)(14,20);(1,4,5,17)(2,8,10,25)(3,12)(6,19,20,18)(7,21,23,24)(9,26,11,16)(13,31,14,29)(15,30)(22,32)(27,28);(1,3)(2,7)(4,12)(5,13)(6,15)(8,21)(9,23)(10,22)(11,27)(14,20)(16,28)(17,31)(18,29)(19,30)(24,26)(25,32)

Polycyclic presentation:

2^10|b^2=g,c^2=eghij,f^2=j,g^2=ij,h^2=j,c^a=cf,c^b=ce,d^b=dg,d^c=dghij,e^a=ei,e^b=eh,e^c=ehij,e^d=ehij,f^a=fj,f^b=fij,f^c=fij,f^e=fj,g^c=ghj,g^d=gij,g^e=gij,h^b=hi,h^c=hi,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339883, #72339887
maximal normal subgroup of	#72339883, #72339887

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