#72239624

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂×#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

3138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,8,5,10,6,9)(3,11,13,7,14,15,12,16)(17,18)(19,27,30,23)(20,33)(21,25)(22,26)(24,35)(28,38)(29,45,34,48)(31,41,32,40)(36,44)(37,43)(39,47,42,46);(1,3,6,12,5,14,4,13)(2,7,8,15,10,16,9,11)(17,19,28,31,22,30,21,32)(18,23,38,40,26,27,25,41)(20,34,36,47,37,29,35,46)(24,42,33,45,44,39,43,48);(3,12,14,13)(7,11,16,15)(17,20,22,37)(18,24,26,44)(19,29)(21,35,28,36)(23,39)(25,43,38,33)(27,42)(30,34)(31,46)(32,47)(40,48)(41,45)

Polycyclic presentation:

2^10|a^2=d,b^2=dei,c^2=eghj,d^2=j,e^2=i,f^2=j,g^2=ij,h^2=j,b^a=bf,c^a=cfgij,c^b=ch,d^b=di,e^a=ej,e^b=ei,f^a=fij,g^a=gj,g^c=gi,h^b=hj,h^c=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239787, #72239914
maximal normal subgroup of	#72239787, #72239914
maximal quotient of	#72239642, #72239879

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