#72338557

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	C₄	#44871
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#777670
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5192 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)(3,12,16,28)(4,18,21,19)(5,7)(6,11)(8,13,14,17)(9,25,27,22)(10,23)(15,24)(20,26)(29,32)(30,31);(2,7,11,10)(3,13,16,17)(9,19,27,18)(15,20,32,31);(1,2)(3,14,16,8)(4,19)(5,7)(6,11)(9,25)(10,23)(12,13,28,17)(15,30,32,26)(18,21)(20,24,31,29)(22,27);(1,3,6,16)(2,8,11,14)(4,20,21,31)(5,13,23,17)(7,12,10,28)(9,26,27,30)(15,22,32,25)(18,24,19,29);(1,4,6,21)(2,9)(3,15)(5,22,23,25)(7,19)(8,24,14,29)(10,18)(11,27)(12,26,28,30)(13,20)(16,32)(17,31)

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=h,d^2=j,e^2=ij,g^2=j,b^a=bg,c^a=cf,c^b=cgj,d^a=dghij,d^b=dgj,d^c=dhi,e^a=egij,e^c=eghj,e^d=eghij,f^d=fj,f^e=fj,g^a=gj,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

106 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, 1, F.6);;
gap> SetPower(coll, 2, F.9);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9*F.10);;
gap> SetPower(coll, 7, 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.7*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, 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, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);