#72321783

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13340	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2333 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efj,c^2=ei,d^2=ef,f^2=j,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=cij,c^b=cfi,d^a=di,d^b=defgj,d^c=defi,e^b=ej,f^a=fj,f^b=fj,g^a=gj,g^c=gj,h^a=hi,h^b=hi,h^c=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321803, #72322750
maximal normal subgroup of	#72321803, #72322750

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