#72251957

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	#35438
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#1117546
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4144 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fhj,h^2=j,c^a=cg,c^b=cf,d^a=dfgi,d^b=dfgi,d^c=dgij,e^a=eg,e^b=egij,e^c=eghi,e^d=ehi,f^d=fj,f^e=fj,g^d=gj,h^c=hj,h^d=hj,h^e=hj,i^c=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251959, #72251962
maximal normal subgroup of	#72251959, #72251962

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