#72302104

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#115133
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, cosocle	C₂³×#41	C₂³
prime radical	#6437623	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=h,c^2=ef,e^2=j,b^a=bd,c^a=ce,c^b=cg,d^c=dh,e^b=ej,e^c=ehj,f^a=fh,f^b=fi,f^c=fhj,f^d=fj,f^e=fj,g^a=ghj,g^c=gij,h^c=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72302105, #72304428, #72304445
maximal normal subgroup of	#72302105, #72304428, #72304445

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