#72306428

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#62477	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5807 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)(4,9)(5,15)(6,16)(7,11)(8,13)(10,14)(17,18)(19,20,25,24)(21,26)(22,35)(23,28)(27,38)(29,31,40,39)(30,36,41,32)(33,44)(34,43)(37,42)(45,48,46,47);(3,10)(5,8)(12,14)(13,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,25)(26,39)(27,36)(31,43)(32,35)(33,45)(34,40)(37,46)(38,41)(42,48)(44,47);(1,3)(2,8)(4,13)(5,11)(6,15)(7,10)(9,14)(12,16)(17,20,23,24)(18,25)(19,28)(21,31,34,39)(22,36,38,32)(26,40)(27,30)(29,43)(33,48,37,47)(35,41)(42,45)(44,46);(1,4,7,6)(2,9,11,16)(3,13,10,15)(5,12,8,14)(17,21,23,34)(18,26,28,43)(19,29,25,40)(20,31,24,39)(22,37,38,33)(27,44,35,42)(30,46,41,45)(32,48,36,47);(1,5,7,8)(2,10,11,3)(4,14,6,12)(9,13,16,15)(17,22,23,38)(18,27,28,35)(19,30,25,41)(20,32,24,36)(21,33,34,37)(26,42,43,44)(29,45,40,46)(31,47,39,48)

Polycyclic presentation:

2^10|a^2=f,c^2=fgi,d^2=ij,e^2=ij,h^2=j,b^a=bg,c^a=cfgh,c^b=cgij,e^a=efhi,e^b=ej,e^c=eij,e^d=eij,f^b=fi,f^c=fi,g^a=gi,g^c=gi,h^a=hj,h^b=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306523, #72306526
maximal normal subgroup of	#72306523, #72306526

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