#72251528

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#110401
commutator	C₂³×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×D₄	C₂³
prime radical	C₂²×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁴×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3265 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)(4,15,18,26)(5,9)(6,10)(7,14)(8,17,24,20)(12,29,16,32)(13,25)(19,23)(21,27)(22,31,28,30);(1,3)(2,7)(4,16,18,12)(5,19,21,13)(6,14)(8,22)(9,25,27,23)(10,11)(15,29,26,32)(17,30)(20,31)(24,28);(1,4,5,20,6,18,21,17)(2,8,9,26,10,24,27,15)(3,12,13,31,14,16,19,30)(7,22,23,29,11,28,25,32)

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=ef,e^2=j,b^a=bj,c^a=ce,c^b=cd,d^a=dh,d^b=di,d^c=dh,e^b=eh,e^c=ej,f^c=fj,g^c=gi,g^e=gj,h^b=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251547, #72251580, #72251610, …
maximal normal subgroup of	#72251547, #72251580, #72251610, …
maximal quotient of	#72262369

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