#72256873

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	#34963
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=g,c^2=g,d^2=gi,g^2=j,i^2=j,b^a=bh,c^a=ch,c^b=ci,d^b=di,d^c=dij,e^a=egi,e^b=ef,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256934
maximal normal subgroup of	#72256934

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