#72285709

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

2836 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fij,d^2=g,e^2=gi,c^a=cfij,c^b=cfj,d^a=dfh,d^b=dfg,d^c=dhi,e^a=ef,e^b=efhj,e^c=eghj,e^d=eij,f^d=fi,f^e=fij,g^a=gij,g^b=gi,g^c=gj,h^b=hij,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	#72286044, #72310940, #72311043, …
maximal normal subgroup of	#72286044, #72310940, #72311043, …

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