#72319062

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10580932	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|c^2=f,d^2=hj,e^2=h,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dhj,d^c=dg,e^a=eh,e^b=ehj,f^d=fj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319103, #72319579, #72319749
maximal normal subgroup of	#72319103, #72319579, #72319749

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