#72240467

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	#4039357
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#419812	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4053 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:

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

Polycyclic presentation:

2^10|b^2=h,d^2=gij,e^2=hj,b^a=bg,c^a=chi,c^b=cf,d^a=dghi,d^b=dg,e^a=ehij,e^b=eh,e^c=eghj,e^d=ei,f^b=fj,f^d=fj,f^e=fj,g^c=gj,g^d=gj,g^e=gj,h^c=hj,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240477, #72251093, #72251302, …
maximal normal subgroup of	#72240477, #72251093, #72251302, …

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