#72287201

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#22436
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#7624082	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #8192 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1252 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:

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

Polycyclic presentation:

2^10|a^2=j,c^2=e,d^2=eghij,e^2=i,f^2=hj,g^2=ij,h^2=j,c^a=cg,c^b=cf,d^a=dg,d^b=dh,d^c=dfghij,e^b=ej,e^d=ej,f^b=fh,f^c=fhj,f^d=fhj,g^a=gij,g^c=gij,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287223, #72287455
maximal normal subgroup of	#72287223, #72287455

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