#72319342

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	#4941495
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3933 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:

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

Polycyclic presentation:

2^10|a^2=g,c^2=gij,d^2=gj,h^2=j,b^a=bj,c^a=cghj,c^b=cghij,d^b=dgj,d^c=df,e^a=eghi,e^b=ei,e^d=egij,f^a=fj,f^d=fj,g^c=gj,h^b=hj,h^c=hj,h^d=hj,h^e=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319501
maximal normal subgroup of	#72319501

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