#72285085

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

2831 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)(3,10,12,7)(4,9)(5,14)(6,8)(11,15,13,16)(17,18)(19,25)(20,31,34,24)(21,35,38,26)(22,27)(23,30)(28,39,32,41)(29,40,36,43)(33,42)(37,45)(44,48)(46,47);(1,3)(2,7)(4,13,5,11)(6,12)(8,10)(9,16,14,15)(17,19)(18,23,27,25)(20,28,34,32)(21,36)(22,30)(24,39)(26,43,35,40)(29,38)(31,41)(33,44,37,46)(42,47)(45,48);(2,8)(4,5)(9,14)(15,16)(17,20)(18,24)(19,28)(21,37)(22,34)(23,39)(25,41)(26,42)(27,31)(29,44)(30,32)(33,38)(35,45)(36,46)(40,48)(43,47);(1,4)(2,9)(3,11,12,13)(5,6)(7,15,10,16)(8,14)(17,19)(18,25)(20,32,34,28)(21,36,38,29)(22,30)(23,27)(24,41,31,39)(26,43,35,40)(33,44)(37,46)(42,48)(45,47);(1,5,6,4)(2,9)(3,11,12,13)(7,16)(8,14)(10,15)(17,21)(18,26,27,35)(19,29,30,36)(20,33,34,37)(22,38)(23,40)(24,42)(25,43)(28,44)(31,45)(32,46)(39,47,41,48)

Polycyclic presentation:

2^10|a^2=f,b^2=fgh,d^2=f,e^2=fhi,b^a=bg,c^a=cghi,c^b=cgj,d^a=df,d^b=dghi,d^c=dhi,e^a=eghj,e^b=efi,e^c=ehi,e^d=ehij,f^b=fi,f^c=fj,f^e=fj,g^a=gi,g^b=gj,g^d=gij,g^e=gi,h^a=hij,h^b=hij,h^d=hj,h^e=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285153, #72285762, #72286199, …
maximal normal subgroup of	#72285153, #72285762, #72286199, …

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