#72334832

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2245234
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂²×#120	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2699 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=hj,b^a=bhi,c^a=cij,c^b=ci,d^a=dh,d^b=dg,d^c=df,e^a=eghij,e^c=ei,e^d=ei,f^a=fj,f^d=fj,g^a=gj,g^b=gj,g^d=gj,h^b=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334833, #72334892, #72334895, …
maximal normal subgroup of	#72334833, #72334892, #72334895, …

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