#72341090

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#35346
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

5843 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:

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

Polycyclic presentation:

2^10|a^2=ij,b^2=f,c^2=ghj,d^2=fghi,e^2=i,b^a=bg,c^a=cgij,c^b=cfgh,d^a=di,d^b=dh,d^c=dfg,e^a=ei,e^b=egij,e^c=ei,e^d=egj,f^a=fi,f^c=fj,f^d=fij,f^e=fi,g^b=gi,g^c=gi,g^d=gi,h^a=hi,h^b=hij,h^c=hi,h^d=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341091, #72341145
maximal normal subgroup of	#72341091, #72341145

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