#72240487

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2747165
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4844 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=gi,d^2=ij,e^2=hi,b^a=bh,c^a=chj,c^b=cfg,d^a=dgh,d^b=dhij,d^c=dhij,e^a=ehj,e^b=eghj,e^c=egh,e^d=eghij,f^c=fj,f^d=fj,f^e=fj,g^b=gj,g^c=gj,g^d=gj,g^e=gj,h^d=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240493, #72251302, #72317935, …
maximal normal subgroup of	#72240493, #72251302, #72317935, …

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