#72287180

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	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623084	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2372 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fhij,e^2=hj,f^2=h,g^2=i,i^2=j,c^a=cj,d^a=dj,d^b=dhj,d^c=dg,e^a=ej,e^b=ehj,e^c=egh,e^d=ei,f^d=fj,f^e=fj,g^c=gi,g^d=gi,g^e=gij,i^c=ij,i^d=ij,i^e=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287278, #72287390, #72320823
maximal normal subgroup of	#72287278, #72287390, #72320823

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