#72287128

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14715
commutator, Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#7656942	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

5716 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=g,d^2=efj,e^2=ij,f^2=ij,h^2=j,b^a=bg,c^a=cg,d^a=de,d^b=defij,d^c=dh,e^a=ei,e^b=ei,e^c=eij,f^a=fj,f^b=fi,f^c=fi,g^d=gj,h^a=hi,h^b=hj,h^c=hj

Conjugacy classes & character table

79 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287165
maximal normal subgroup of	#72287165

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