#72237086

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2077561
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=fghj,c^2=fgj,g^2=j,h^2=j,b^a=bf,c^a=cfgh,c^b=cgi,d^a=dij,d^b=dgh,d^c=dij,e^a=egj,e^b=egh,e^c=egi,f^a=fj,g^b=gj,g^e=gj,h^a=hj,h^b=hj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237665, #72251715, #72251931
maximal normal subgroup of	#72237665, #72251715, #72251931

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