#72321891

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	C₂⁵	#137
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=egij,h^2=i,i^2=j,b^a=bhij,c^a=cg,c^b=cef,d^a=dghj,d^b=defj,d^c=de,e^a=ej,g^b=gj,g^d=gj,h^b=hi,h^d=hij,i^b=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321926, #72322691
maximal normal subgroup of	#72321926, #72322691

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