#72339680

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1839160
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4460 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)(4,10)(5,6)(7,24)(8,11)(9,15)(13,26)(14,22)(16,27)(17,25)(18,20)(19,32)(21,23)(28,29)(30,31);(1,3)(2,9)(4,17)(5,22)(6,16)(7,14)(8,15)(10,26)(11,27)(12,24)(13,20)(18,31)(19,29)(21,30)(23,28)(25,32);(1,4,6,21)(2,10)(3,13,16,29)(5,23)(7,19,8,20)(9,25)(11,18)(12,28)(14,17,15,30)(22,31)(24,32)(26,27);(1,5,6,2)(3,9,16,22)(4,18,21,32)(7,24,8,11)(10,19,23,20)(12,14,27,15)(13,26,29,28)(17,31,30,25);(3,14,16,15)(9,12,22,27)(13,17,29,30)(25,28,31,26)

Polycyclic presentation:

2^10|c^2=ghj,d^2=j,e^2=i,g^2=j,h^2=j,b^a=bg,c^a=cf,c^b=cfgij,d^a=dj,d^b=dfi,d^c=dh,e^b=ehj,e^d=ei,f^b=fj,f^c=fj,g^a=gj,g^b=gj,h^b=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

76 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, 3, F.7*F.8*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, 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);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*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.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*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, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);