#72284019

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

2972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=i,c^2=f,d^2=hi,b^a=bij,c^a=ci,c^b=ch,d^a=dh,d^b=dij,d^c=dg,e^a=ef,e^b=egij,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

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72285163, #72285517, …
maximal normal subgroup of	#72284709, #72285163, #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);;
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);;
gap> SetConjugate(coll, 4, 2, F.4*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9*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);