#72310895

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	#245249	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=fghi,e^2=fgij,b^a=bg,c^a=cgh,c^b=cfg,d^a=di,d^b=di,d^c=dfj,e^a=ehj,e^b=ehi,e^c=egh,e^d=efg,f^a=fi,f^c=fi,f^e=fj,g^b=gi,g^c=gj,g^e=gj,h^b=hj,h^c=hij,h^d=hi,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310940, #72340295, #72340401, …
maximal normal subgroup of	#72310940, #72340295, #72340401, …

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