#72340282

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	#245255	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2779 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=ghj,e^2=j,b^a=bg,c^a=cgh,c^b=cfgi,d^a=di,d^b=di,d^c=dfi,e^b=ehj,e^d=egij,f^a=fi,f^c=fi,f^e=fj,g^b=gi,g^c=gj,h^b=hj,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340288, #72340307, #72340309, …
maximal normal subgroup of	#72340288, #72340307, #72340309, …

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