#72285060

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	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233549	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3214 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)(3,6)(4,12)(5,8)(7,14)(9,11)(10,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,35)(28,41)(29,37)(30,38)(31,39)(32,40)(33,36)(34,42)(43,47)(44,46)(45,48);(1,2)(3,9,11,6)(4,12,14,7)(5,8)(10,16)(13,15)(17,19)(18,23)(20,30)(21,33,34,28)(22,29)(24,38)(25,41,42,36)(26,37)(27,32)(31,44,45,43)(35,40)(39,47,48,46);(1,3)(2,6)(4,13,14,10)(5,11)(7,15,12,16)(8,9)(17,20)(18,24)(19,27,29,30)(21,31,34,45)(22,32)(23,35,37,38)(25,39,42,48)(26,40)(28,43)(33,44)(36,46)(41,47);(1,4,5,14)(2,7,8,12)(3,10)(6,15)(9,16)(11,13)(17,21)(18,25)(19,28,29,33)(20,31)(22,34)(23,36,37,41)(24,39)(26,42)(27,43,30,44)(32,45)(35,46,38,47)(40,48);(2,8)(4,14)(6,9)(10,13)(18,26)(19,29)(24,40)(25,42)(27,30)(28,33)(39,48)(43,44)

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=fhj,b^a=bh,c^b=cg,d^a=dfhj,d^b=dfghi,d^c=dghij,e^a=eij,e^b=eij,e^d=ej,f^b=fi,f^d=fij,g^a=gj,g^b=gj,g^c=gi,g^d=gi,h^c=hj,h^d=hij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285517, #72340377, #72340381, …
maximal normal subgroup of	#72285517, #72340377, #72340381, …

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