#72340243

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	#245246	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2909 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=gi,e^2=ij,b^a=bi,c^a=cf,c^b=cfg,d^a=dh,d^b=dghij,d^c=di,e^a=ehj,e^b=efg,e^c=ehij,e^d=efgi,f^b=fi,f^c=fi,f^d=fj,f^e=fj,g^a=gi,g^b=gi,g^e=gj,h^b=hi,h^c=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340248, #72340293, #72340311, …
maximal normal subgroup of	#72340248, #72340293, #72340311, …

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