#72284653

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

3265 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=f,d^2=fhij,b^a=bh,c^a=ch,c^b=cg,d^a=dfg,d^b=dfgh,d^c=dgh,e^a=ej,e^b=eij,e^d=ej,f^a=fj,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	#72284709, #72340359, #72340364, …
maximal normal subgroup of	#72284709, #72340359, #72340364, …

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.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*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, 1, F.6*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);