#72285686

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

2972 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:

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

Polycyclic presentation:

2^10|c^2=fgij,e^2=ghi,b^a=bf,c^a=cfg,c^b=ch,d^a=df,d^c=dfi,e^a=ef,e^b=efh,e^c=efghj,e^d=eh,f^c=fj,f^e=fi,g^b=gij,g^c=gj,g^d=gj,g^e=gj,h^a=hi,h^c=hij,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310975, #72340222, #72340239, …
maximal normal subgroup of	#72310975, #72340222, #72340239, …

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