#72257487

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1438
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#10575509	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=fj,c^2=efgh,e^2=i,f^2=j,h^2=j,b^a=bf,c^a=cfg,c^b=ce,d^a=dj,d^b=dj,d^c=dij,e^a=ei,e^b=ej,e^c=ej,f^a=fj,f^c=fij,g^b=gj,g^c=gj,h^b=hij,h^c=hij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257538
maximal normal subgroup of	#72257538
maximal quotient of	#72257714

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