#72285804

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

2797 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=f,d^2=ghj,e^2=ghi,b^a=bi,c^a=cf,c^b=ci,d^a=dfh,d^b=dfg,d^c=dhi,e^a=efij,e^b=efij,e^c=eg,e^d=eij,f^d=fi,f^e=fij,g^a=gij,g^c=gij,g^d=gj,g^e=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, #72340222, #72340248, …
maximal normal subgroup of	#72286044, #72340222, #72340248, …

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