#72306374

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1390
commutator	C₂²×Q₈	C₂×C₄²
Frattini, cosocle	C₂²×#124	C₂³
prime radical	C₂²×#532	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂⁴×C₄ ◁ C₂²×#532 ◁ G
LECS	G ▷ C₂²×#124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=d,c^2=dfj,d^2=i,f^2=j,g^2=j,h^2=j,b^a=bij,c^a=cg,c^b=ce,d^c=dh,e^c=ej,f^a=fhj,f^b=fj,f^c=fh,f^d=fij,g^a=gh,g^b=gj,g^c=ghj,g^d=gij,h^a=hi,h^c=hi,h^f=hj,h^g=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306376, #72306381, #72306522
maximal normal subgroup of	#72306376, #72306381, #72306522

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