#72250428

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32554
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(6,11,14,27)(8,20)(10,24)(12,25,28,15)(13,26,31,30)(16,33,32,29)(18,35)(19,34)(21,23)(22,36);(2,3)(5,6,9,14)(7,16,17,32)(8,12,23,28)(10,25,24,15)(11,20,27,21)(13,19,31,34)(18,26,22,30)(29,36,33,35);(1,2)(3,4)(6,12)(7,17)(8,21)(10,24)(11,25)(13,29)(14,28)(15,27)(16,26)(18,35)(20,23)(22,36)(30,32)(31,33);(1,3,4,2)(5,7,10,19,9,17,24,34)(6,13,25,16,14,31,15,32)(8,22,20,36,23,18,21,35)(11,26,12,29,27,30,28,33)

Polycyclic presentation:

2^10|a^2=h,b^2=j,d^2=fghij,e^2=i,i^2=j,b^a=be,c^b=cef,d^a=dg,d^b=dfi,d^c=dfh,e^a=ei,e^b=eij,e^c=eij,e^d=ej,f^d=fj,g^d=gj,h^b=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250490, #72250807, #72257203, …
maximal normal subgroup of	#72250490, #72250807, #72257203, …

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