#72313179

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24404
commutator, Frattini, cosocle	C₂²×#41	C₂⁴
LCS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5552 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:

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

Polycyclic presentation:

2^10|c^2=egj,d^2=efgij,g^2=j,h^2=j,c^a=ce,c^b=cg,d^a=def,d^b=dg,d^c=dfhj,e^d=ei,f^c=fj,f^d=fi,g^b=gj,h^a=hi,h^c=hij,h^d=hi,h^e=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313185, #72337432, #72337466
maximal normal subgroup of	#72313185, #72337432, #72337466

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