#72287604

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23001
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#19603	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8293 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

747 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=i,d^2=gij,e^2=i,f^2=h,g^2=ij,h^2=j,b^a=bj,c^a=cj,c^b=cgi,d^a=df,d^b=di,d^c=dfghij,e^d=ej,f^a=fh,f^c=fhj,f^d=fhj,g^c=gij,h^a=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287651
maximal normal subgroup of	#72287651

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