#72338554

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3334833
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5274 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gj,d^2=gij,e^2=ij,f^2=j,g^2=j,b^a=bf,c^a=cfgj,c^b=cgh,d^a=dghi,d^b=di,d^c=dh,e^a=ehij,e^b=egij,e^d=egj,f^a=fj,f^c=fj,f^e=fj,g^a=gj,g^d=gj,h^b=hj,h^c=hj,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338568
maximal normal subgroup of	#72338568

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