#72334831

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	#3381329
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

2636 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)(4,8,15,23)(5,9)(6,10)(11,18,28,21)(12,19)(13,20)(14,25,29,22)(16,24)(17,26)(27,32,30,31);(1,3)(2,7)(4,11)(5,16,17,12)(6,13)(8,21)(9,24,26,19)(10,20)(14,27,29,30)(15,28)(18,23)(22,32,25,31);(1,4)(2,8,10,23)(3,11)(5,14,17,29)(6,15)(7,18,20,21)(9,25)(12,27,16,30)(13,28)(19,32)(22,26)(24,31);(1,5)(2,9)(3,12)(4,14,15,29)(6,17)(7,19)(8,22,23,25)(10,26)(11,27,28,30)(13,16)(18,31,21,32)(20,24);(7,20)(9,26)(18,21)(22,25)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=fi,d^2=h,b^a=bh,c^a=chi,c^b=cfi,d^a=dh,d^b=dfg,d^c=dfhi,e^a=egh,e^b=ei,e^d=ei,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334833, #72334896, #72334899, …
maximal normal subgroup of	#72334833, #72334896, #72334899, …

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