#72285968

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

2799 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgj,d^2=ghj,e^2=gj,b^a=bg,c^a=cfgh,c^b=cfghi,d^a=dfhj,d^b=dhij,d^c=dfghj,e^b=efhij,e^c=eghj,e^d=ehj,f^a=fi,f^c=fj,f^d=fij,f^e=fij,g^b=gi,g^c=gj,g^d=gj,h^a=hi,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	#72340222, #72340246, #72340293, …
maximal normal subgroup of	#72340222, #72340246, #72340293, …

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