#72321988

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3414	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2036 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=ef,d^2=eghij,e^2=j,g^2=i,h^2=ij,i^2=j,b^a=bh,c^a=ceghj,c^b=ceghij,d^a=defghij,d^b=defghi,d^c=deghi,e^a=ej,e^c=ej,e^d=ej,f^c=fj,f^d=fj,g^b=gi,g^d=gj,h^b=hij,h^c=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322059, #72322784
maximal normal subgroup of	#72322059, #72322784

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