#72285966

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2967 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)(3,9,12,6)(4,7)(5,8)(10,15,11,14)(13,16)(17,18)(19,27,28,23)(20,24)(21,34,37,25)(22,26)(29,38,31,42)(30,43)(32,46,36,41)(33,40)(35,39)(44,47)(45,48);(1,3)(2,6)(4,10)(5,12)(7,15)(8,9)(11,13)(14,16)(19,28)(23,27)(24,40)(30,35)(38,42)(39,43)(41,46)(47,48);(1,4)(2,7,8,16)(3,10,12,11)(5,13)(6,14)(9,15)(17,19)(18,23,26,27)(20,31,33,29)(21,35,37,30)(22,28)(24,38)(25,39)(32,45)(34,43)(36,44)(40,42)(41,48,46,47);(1,4)(2,7)(3,11,12,10)(5,13)(6,14,9,15)(8,16)(17,20)(18,24)(19,29,28,31)(21,36,37,32)(22,33)(23,38,27,42)(25,41,34,46)(26,40)(30,44)(35,45)(39,48)(43,47);(7,16)(14,15)(17,21)(18,25)(19,30)(20,32)(22,37)(23,39)(24,41)(26,34)(27,43)(28,35)(29,44)(31,45)(33,36)(38,47)(40,46)(42,48)

Polycyclic presentation:

2^10|a^2=f,c^2=fhij,d^2=f,b^a=bg,c^a=cgh,c^b=cgij,d^a=df,d^b=dghij,d^c=dhi,e^a=eghij,e^c=ehij,e^d=ehij,f^b=fi,f^c=fj,f^e=fj,g^a=gi,g^c=gi,g^d=gij,h^a=hij,h^c=hj,h^d=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285984, #72310982, #72311005, …
maximal normal subgroup of	#72285984, #72310982, #72311005, …

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