#72321173

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3184
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	#23761
prime radical	C₂²×#135	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1348 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ef,d^2=ghi,e^2=j,g^2=i,h^2=i,i^2=j,b^a=bh,c^a=cfgij,d^a=degj,d^b=dghij,d^c=df,f^a=fj,g^a=gi,g^b=gi,g^c=gij,g^d=gij,h^a=hij,h^b=hi,h^c=hij,h^d=hij,i^a=ij,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321276, #72322569
maximal normal subgroup of	#72321276, #72322569

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