#72257367

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#393
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₂×#1439
prime radical	C₂³×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3831 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=ehj,e^2=i,f^2=i,h^2=j,b^a=bf,c^a=ce,c^b=cfg,d^b=di,d^c=di,e^a=ei,e^b=eij,e^c=ei,f^a=fi,f^c=fij,g^a=gi,g^c=gi,h^a=hi,h^c=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257390
maximal normal subgroup of	#72257390
maximal quotient of	#72257633

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