#72250405

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13334	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2228 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=j,d^2=egh,e^2=ij,f^2=j,i^2=j,c^a=ce,c^b=cefj,d^a=degij,d^b=degh,d^c=deghij,e^a=ei,e^b=eij,e^c=ei,e^d=ej,f^a=fj,f^b=fj,f^c=fj,f^d=fj,g^c=gj,g^d=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250436, #72250497, #72250983, …
maximal normal subgroup of	#72250436, #72250497, #72250983, …

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