#72237871

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	#4866548
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5383 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)(3,11,14,7)(4,15,17,8)(5,18,21,9)(6,10)(12,22)(13,23)(16,29)(19,26)(20,24)(25,28)(27,31,30,32);(1,3)(2,7)(4,12)(5,19)(6,14)(8,22)(9,23)(10,11)(13,21)(15,25)(16,30)(17,28)(18,26)(20,27)(24,31)(29,32);(7,11)(22,25)(23,26)(31,32);(1,4)(2,8)(3,12,14,28)(5,20)(6,17)(7,22,11,25)(9,24)(10,15)(13,27,19,30)(16,21)(18,29)(23,32,26,31);(1,5,6,21)(2,9)(3,13,14,19)(4,16,17,20)(7,23)(8,24)(10,18)(11,26)(12,27,28,30)(15,29)(22,31)(25,32)

Polycyclic presentation:

2^10|a^2=g,d^2=h,e^2=hij,b^a=bf,c^a=ch,c^b=chi,d^a=dhi,d^b=dghj,e^a=egij,e^b=ehij,e^d=eij,f^a=fj,f^e=fj,g^b=gj,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237873, #72237885, #72237907, …
maximal normal subgroup of	#72237873, #72237885, #72237907, …

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