#72319545

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623099	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2382 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)(3,12)(4,15)(5,16)(6,9)(7,11)(8,13)(10,14)(17,18)(19,30)(20,27)(21,26)(22,28)(23,29)(24,31)(25,37)(32,41)(33,46)(34,47)(35,40)(36,45)(38,44)(39,42)(43,48);(3,13)(5,14)(8,12)(10,16)(19,31)(24,30)(32,35)(33,39)(34,48)(40,41)(42,46)(43,47);(1,3,6,5,7,13,4,14)(2,8,9,10,11,12,15,16)(17,19,21,33,22,31,23,39)(18,24,26,42,28,30,29,46)(20,35,36,48,37,32,38,34)(25,40,44,43,27,41,45,47);(1,4,7,6)(2,9,11,15)(3,14,13,5)(8,10,12,16)(19,32)(20,36)(21,23)(24,40)(25,44)(26,29)(27,45)(30,41)(31,35)(33,48)(34,39)(37,38)(42,47)(43,46);(1,5,6,13,7,14,4,3)(2,10,9,12,11,16,15,8)(17,20,21,36,22,37,23,38)(18,25,26,44,28,27,29,45)(19,32,33,34,31,35,39,48)(24,41,42,47,30,40,46,43)

Polycyclic presentation:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319621, #72319631, #72319699
maximal normal subgroup of	#72319621, #72319631, #72319699

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