#72285441

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

2785 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)(3,9,11,6)(4,12)(5,8)(7,13)(10,14,16,15)(17,18)(19,27,30,23)(20,31)(21,35)(22,26)(24,34)(25,36)(28,40,32,38)(29,43,45,39)(33,41)(37,42)(44,48,46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(21,36)(24,31)(25,35)(29,45)(38,40)(39,43)(41,42)(47,48);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16)(17,19)(18,23)(20,32,34,28)(21,29)(22,30)(24,40,31,38)(25,39)(26,27)(33,46,37,44)(35,43)(36,45)(41,48,42,47);(4,13)(6,9)(7,12)(14,15)(17,20)(18,24)(19,28)(21,37)(22,34)(23,38)(25,42)(26,31)(27,40)(29,46)(30,32)(33,36)(35,41)(39,48)(43,47)(44,45);(3,11)(4,13)(7,12)(14,15)(17,21,22,36)(18,25,26,35)(19,29,30,45)(20,33,34,37)(23,39,27,43)(24,41,31,42)(28,44,32,46)(38,47,40,48)

Polycyclic presentation:

2^10|a^2=f,c^2=h,e^2=i,b^a=bh,c^a=cg,d^a=dfi,d^b=dgh,d^c=dfj,e^a=efi,e^b=ehij,e^c=ej,e^d=ei,f^b=fj,f^c=fi,g^a=gi,g^c=gj,g^d=gi,g^e=gi,h^a=hj,h^d=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286185
maximal normal subgroup of	#72286185

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