#72311083

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#35346	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=fj,c^2=g,d^2=gij,b^a=bf,c^a=ch,c^b=chj,d^a=dfhi,d^b=dfgh,d^c=dg,e^b=ej,e^c=efi,e^d=ehij,f^c=fj,f^d=fi,g^a=gij,g^b=gij,g^e=gj,h^b=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311155, #72340305, #72340327, …
maximal normal subgroup of	#72311155, #72340305, #72340327, …

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