#72265628

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1762
commutator	C₂×C₈	C₂²×C₄²
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#60673	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×C₄×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=fhj,d^2=ij,e^2=h,f^2=j,h^2=j,i^2=j,b^a=be,c^b=ci,d^a=dh,d^b=dh,d^c=dj,e^a=ei,e^b=eh,e^c=ej,f^a=fj,g^b=ghi,h^a=hj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265638, #72265703, #72265712, …
maximal normal subgroup of	#72265638, #72265703, #72265712, …

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