#72334824

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#4614842
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

2961 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)(4,14,16,8)(5,9)(6,10)(11,21,26,18)(12,19)(13,20)(15,22,29,23)(17,24)(25,30,28,31)(27,32);(1,3,6,13)(2,7,10,20)(4,11,16,26)(5,12)(8,21,14,18)(9,19)(15,28)(17,27)(22,30)(23,31)(24,32)(25,29);(1,4,6,16)(2,8,10,14)(3,11,13,26)(5,15)(7,18,20,21)(9,23)(12,25)(17,29)(19,31)(22,24)(27,28)(30,32);(1,5)(2,9)(3,12)(4,15,16,29)(6,17)(7,19)(8,22,14,23)(10,24)(11,25,26,28)(13,27)(18,30,21,31)(20,32);(3,13)(5,17)(11,26)(15,29)

Polycyclic presentation:

2^10|a^2=h,b^2=f,c^2=f,d^2=h,b^a=bh,c^a=ci,c^b=cfij,d^a=dh,d^b=dfg,d^c=dfhij,e^a=egh,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

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334827, #72334902, #72334916, …
maximal normal subgroup of	#72334827, #72334902, #72334916, …

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