#72321727

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23573
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3060	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2243 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=efhij,d^2=efhj,e^2=j,f^2=i,g^2=i,h^2=ij,i^2=j,b^a=bf,c^a=cfgij,c^b=cefg,d^a=di,e^a=ej,f^b=fi,f^d=fj,g^b=gij,g^c=gj,g^d=gj,h^b=hij,h^d=hj,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321803, #72322519
maximal normal subgroup of	#72321803, #72322519

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