#72284077

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

2776 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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=f,d^2=hij,e^2=i,b^a=bij,c^a=ci,c^b=ch,d^a=dhi,d^c=dg,e^a=efi,e^b=egj,e^d=ef,f^b=fj,f^d=fi,g^a=gj,g^c=gi,g^d=gj,h^c=hj,h^e=hi

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285517
maximal normal subgroup of	#72285517

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