#72250408

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24338
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

2309 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gi,d^2=ehij,e^2=i,f^2=j,i^2=j,b^a=be,c^b=ceh,d^a=defj,d^b=defghj,d^c=dehj,e^a=ei,e^b=ei,e^c=eij,e^d=ej,f^a=fj,f^c=fj,f^d=fj,g^b=gj,g^d=gj,h^d=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250436, #72250497, #72252629, …
maximal normal subgroup of	#72250436, #72250497, #72252629, …

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