#72320609

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33595	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3283 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fj,d^2=gh,e^2=i,f^2=j,g^2=i,h^2=ij,i^2=j,b^a=bh,c^a=cghi,c^b=cj,d^a=dfghij,d^b=dgh,d^c=dgh,e^b=ei,e^c=ej,f^a=fj,g^b=gi,g^c=gj,g^d=gj,h^b=hij,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320850, #72321960
maximal normal subgroup of	#72320850, #72321960

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