#72339679

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2319841
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5823 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)(3,12)(4,9)(5,10)(6,23)(7,11)(8,16)(13,26)(14,15)(17,25)(18,24)(19,20)(21,28)(22,27)(29,31)(30,32);(1,3)(2,8)(4,18)(5,17)(6,14)(7,16)(9,26)(10,15)(11,25)(12,23)(13,21)(19,24)(20,31)(22,30)(27,29)(28,32);(1,4,5,22)(2,9)(3,13,17,31)(6,20,7,21)(8,24)(10,27)(11,28)(12,29)(14,18,16,30)(15,32)(19,23)(25,26);(2,10)(3,14,17,16)(8,25,15,12)(9,27)(11,23)(13,30,31,18)(19,28)(24,29,32,26);(1,2)(3,15)(4,19)(5,10)(6,11)(7,23)(8,17)(9,20)(12,14)(13,26)(16,25)(18,32)(21,27)(22,28)(24,30)(29,31)

Polycyclic presentation:

2^10|c^2=ghj,d^2=i,g^2=j,h^2=j,b^a=bg,c^a=cf,c^b=cfgij,d^a=dj,d^b=dfhi,d^c=di,e^b=ej,e^c=ehi,e^d=eij,f^b=fj,f^c=fj,g^a=gj,g^b=gj,h^b=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339689
maximal normal subgroup of	#72339689

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