#72334817

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	#4328155
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#2021	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3016 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)(3,7,13,20)(4,14)(5,9,17,23)(8,16)(11,24)(12,27,28,19)(15,21)(18,26)(22,30)(25,32)(29,31);(1,3)(2,7)(4,11)(5,12,17,28)(6,13)(8,18)(9,19,23,27)(10,20)(14,24)(15,29,30,25)(16,26)(21,32,22,31);(1,4)(2,8)(3,11)(5,15,17,30)(6,16)(7,18)(9,22,23,21)(10,14)(12,25,28,29)(13,26)(19,32,27,31)(20,24);(1,5,6,17)(2,9,10,23)(3,12,13,28)(4,15)(7,19,20,27)(8,21)(11,25)(14,22)(16,30)(18,31)(24,32)(26,29);(3,13)(5,17)(11,26)(15,30)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=f,d^2=h,b^a=bf,c^a=ci,c^b=cf,d^a=dfg,d^b=dfgj,d^c=dfhij,e^a=eghj,e^b=eij,e^d=eij,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334823, #72334842, #72334872, …
maximal normal subgroup of	#72334823, #72334842, #72334872, …

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