#72340223

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

2796 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=g,e^2=fj,b^a=bi,c^a=cf,c^b=cfg,d^a=dh,d^b=dghj,d^c=di,e^a=eij,e^b=efh,e^c=ehi,e^d=efgj,f^b=fi,f^c=fi,f^d=fj,g^a=gi,g^b=gi,g^e=gj,h^b=hi,h^c=hj,h^d=hi,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340228, #72340301, #72340316, …
maximal normal subgroup of	#72340228, #72340301, #72340316, …

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