#72285054

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

2838 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:

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

Polycyclic presentation:

2^10|c^2=f,d^2=fgj,e^2=i,c^a=ch,c^b=ch,d^a=dfg,d^b=dg,d^c=dfhj,e^a=ei,e^b=ehij,e^c=efghij,e^d=eij,f^a=fj,f^b=fj,f^d=fj,f^e=fi,g^a=gj,g^c=gij,g^d=gj,g^e=gi,h^c=hj,h^d=hi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285526, #72286185, #72340309, …
maximal normal subgroup of	#72285526, #72286185, #72340309, …

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