#72339677

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	#2594065
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#394098	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4042 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)(3,12)(4,16)(5,22)(6,11)(7,10)(8,14)(9,19)(13,27)(15,25)(17,24)(18,21)(20,28)(23,26)(29,32)(30,31);(1,3)(2,8)(4,17,19,29)(5,23)(6,14)(7,15)(9,27,16,31)(10,26)(11,25)(12,22)(13,21,30,20)(18,32,28,24);(1,4,7,19)(2,9)(3,13)(5,20,6,21)(8,24,26,32)(10,16)(11,28)(12,27,25,31)(14,17)(15,30)(18,22)(23,29);(2,10)(3,14,15,23)(8,12,26,25)(9,16)(11,22)(13,29,30,17)(18,28)(24,31,32,27);(1,2)(3,8)(4,18)(5,11)(6,22)(7,10)(9,20)(12,14)(13,27)(15,26)(16,21)(17,32)(19,28)(23,25)(24,29)(30,31)

Polycyclic presentation:

2^10|b^2=f,c^2=ghj,d^2=i,g^2=j,h^2=j,b^a=bg,c^a=cfi,c^b=cgij,d^a=dj,d^b=dfhi,d^c=di,e^c=eh,e^d=eij,f^c=fj,g^a=gj,g^b=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339689
maximal normal subgroup of	#72339689

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