#72284554

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
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2798 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:

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

Polycyclic presentation:

2^10|b^2=fi,d^2=gj,e^2=g,b^a=bf,c^a=cf,c^b=cj,d^a=dfh,d^b=dfg,d^c=di,e^a=efij,e^b=efghij,e^c=egij,f^d=fi,f^e=fij,g^a=gij,g^b=gi,h^b=hj,h^c=hi,h^d=hj,h^e=hj

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284848, #72340222, #72340271, …
maximal normal subgroup of	#72284848, #72340222, #72340271, …

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