#72249886

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	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2961 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2244 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=f,d^2=eghij,e^2=j,g^2=i,i^2=j,b^a=bh,c^a=cf,d^a=de,d^b=dg,d^c=dfij,e^b=ej,e^c=ej,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249954, #72250049, #72250330
maximal normal subgroup of	#72249954, #72250049, #72250330
maximal quotient of	#72307166

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