#72250355

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	#24338
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×D₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1869 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)(7,17,23,36)(8,25)(9,10)(13,28,26,15)(18,24,21,22)(19,29)(20,27)(31,32)(33,34);(1,2)(3,4)(6,13)(7,18)(8,20)(9,10)(12,28)(15,30)(16,26)(17,24)(19,32)(21,23)(22,36)(25,27)(29,31)(33,34);(5,6,10,28,11,16,9,15)(7,19,24,32,23,31,22,29)(8,26,25,12,27,13,20,30)(14,21,34,17,35,18,33,36);(2,4)(5,7,20,36,9,24,27,21,11,23,25,17,10,22,8,18)(6,14,13,31,28,33,30,32,16,35,26,19,15,34,12,29)

Polycyclic presentation:

2^10|a^2=g,c^2=gi,d^2=ehj,e^2=i,f^2=j,h^2=j,i^2=j,c^a=ce,c^b=cehj,d^a=defj,d^b=defgh,d^c=deh,e^a=ei,e^b=eij,e^c=ei,e^d=ej,f^a=fj,f^b=fj,f^d=fj,g^c=gj,g^d=gj,h^a=hj,h^b=hj,h^c=hj,h^d=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	#72250398, #72252637, #72252852
maximal normal subgroup of	#72250398, #72252637, #72252852

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