#72334830

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	#2832567
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

2679 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)(3,7,13,22)(4,14)(5,9)(6,10)(8,16)(11,20,29,23)(12,21,18,26)(15,27)(17,25)(19,24)(28,31,30,32);(5,17)(9,25)(12,18)(21,26);(1,3)(2,7,10,22)(4,11)(5,18,17,12)(6,13)(8,23,14,20)(9,26)(15,30,19,28)(16,29)(21,25)(24,31)(27,32);(1,4)(2,8,10,14)(3,11)(5,19)(6,16)(7,20,22,23)(9,27,25,24)(12,28)(13,29)(15,17)(18,30)(21,31,26,32);(1,5)(2,9)(3,12)(4,15)(6,17)(7,21)(8,24)(10,25)(11,28)(13,18)(14,27)(16,19)(20,31)(22,26)(23,32)(29,30)

Polycyclic presentation:

2^10|a^2=h,c^2=f,d^2=i,c^a=chi,d^a=dg,d^b=dfi,d^c=df,e^b=eghj,e^c=eh,e^d=ehj,f^a=fj,f^e=fj,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	#72334833, #72334872, #72334887, …
maximal normal subgroup of	#72334833, #72334872, #72334887, …

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