#72265643

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	C₂×#1879
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#7623096	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1770 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1993 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=eij,d^2=i,e^2=hi,i^2=j,b^a=bg,c^a=cgij,c^b=ce,d^a=dj,d^b=dhij,d^c=dj,e^a=eh,e^b=ehij,f^c=fj,g^c=ghj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265668, #72265717, #72308521
maximal normal subgroup of	#72265668, #72265717, #72308521

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